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PUBLISHED  BY 

JOHN  WILEY  &  SONS,  Inc.,  NEW  YORK.     | 

CHAPMAN  &  HALL,  Limited,  LONDON.                1 

MATHEMATICAL    MONOGRAPHS 

EDITED   BY 

MANSFIELD  MERRIMAN  and  ROBERT  S.  WOODWARD 


No.  19 


EMPIRICAL    FORMULAS 


BY 

THEODORE   R.    RUNNING 

Associate  Professor  of  Mathematics,  University  of  Michigan 


FIRST    EDITION 


NEW  YORK 

JOHN  WILEY   &   SONS,  Inc. 

London:    CHAPMAN   &   HALL,   Limited 

1917 


Copyright,  19 17, 

BY 

THEODORE   R.   RUNNING 


PRESS  or 

BRAUNWORTH    &    CO. 

BOOK    MANUrACTURCRS 

aoOOKLVN.    N.     Y. 


PREFACE 


This  book  is  the  result  of  an  attempt  to  answer  a  number 
of  questions  which  frequently  confront  engineers.  So  far  as 
the  author  is  aware  no  other  book  in  English  covers  the  same 
groi;iid  in  an  elementary  manner. 

It  is  thought  that  the  method  of  determining  the  constants 
in  formulas  by  the  use  of  the  straight  line  alone  leaves  little  to 
be  desired  from  the  point  of  view  of  simplicity.  The  approxi- 
mation by  this  method  is  close  enough  for  most  problems  arising 
in  engineering  work.  Even  when  the  Method  of  Least  Squares 
must  be  employed  the  process  gives  a  convenient  way  of  obtain- 
ing approximate  values. 

For  valuable  suggestions  and  criticisms  the  author  here 
expresses  his  thanks  to  Professors  Alexander  Ziwet  and  Horace 
W.  King. 

T.  R.  R. 

University  of  Michigan,  191 7. 


366666 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from  j 

Microsoft  Corporation  j 


http://www.archive.org/details/empiricalformulaOOrunnrich 


CONTENTS 


PAGE 

Introduction 9 

CHAPTER  I 

I.  y=a-\-bx-\-cx^-{-dx^-\-  .  .  .   +^x" 13 

Values  of  x  form  an  arithmetical  series  and  A^y  constant. 

II.  y  =  a+^+iL+4+    .    .    .    +1 „ 

X    x^    x^  x" 

Values  of  -  form  an  arithmetical  series  and  A^y  constant. 

X 

III.  -  =  a-{-bx+cx^+dx^+  .  .  .   +gx'' 25 

y 

Values  of  x  form  an  arithmetical  series  and  A"-  constant. 

y 

IV.  y^  =  a-^bx-{-cx^-^dx^-{-  .  .  .   -i-qx^ 25 

Values  of  x  form  an  arithmetical  series  and  A^y"^  constant. 

CHAPTER  II 

V.  y  =  ab^ 27 

Values  of  x  form  an  arithmetical  series  and  the  values  of  y 
form  a  geometrical  series. 

VI.  y  =  a-\-bc^. 28 

Values  of  x  form  an  arithmetical  series  and  the  values  of  Ay 
form  a  geometrical  series. 

VII.  log  y  =  a-^b(f 32 

Values  of  x  form  an  arithmetical  series  and  the  values  of 
A  log  y  form  a  geometrical  series. 

VLll.  y=a-{-bx-{-cd'' S3 

Values  of  x  form  an  arithmetical  series  and  the  values   of 
A^y  form  a  geometrical  series. 
IX.  y  =  io(^  +  f^^  +  cx2 ^^ 

Values  of  x  form  an  arithmetical   series  and  the  values    of 
A2  log  y  constant. 

5 


6  CONTENTS 

PAGE 

X.y^ks'^' 37 

Values  of  x  form  an  arithmetical  series  and  the  values 
A^logy  form  a  geometrical  series. 

"^■y^^nk^' 3« 

Values  of  x  form  an  arithmetical  series  and  A^-  constant. 

y 

CHAPTER  III 

XII.  y=ac(^ 42 

Values  of  x  form  a  geometrical  series  and  the  values  of  y 
form  a  geometrical  series. 

XIII.  y  =a-{-b  log  x-\-c  log^^ 44 

Values  of  x  form  a  geometrical  series  and  A^y  constant. 

XIV.  y=a+bxf' 45 

Values  of  x  form  a  geometrical  series  and  the  values  of  Ay 

form  a  geometrical  series. 

XV.  y=aio^^ ..-.     49 

Values  of  x  form  a  geometrical  series  and  the  values  of 

A  log  y  form  a  geometrical  series. 

CHAPTER  IV 

XVI.  {x+a){y+b)  =c 53 

Point**fepresented  by  Ix-xk,  ^~^j  lie  on  a  straight  line. 

ft 
XVIa.  y=oio*+'= 56 

Points  represented  by  ( log  — ,  log  — )  lie  on  a 

\x—xk        y  ykj 

straight  line. 

XVn.  y^ae'^i-bef^ 58 

Values  of  x  form  an  arithmetical  series  and  the  points  repre- 
sented by  I  ^^S  ^^^±^ )  lie  on  a  straight  line  whose  slope, 

\  yt     yk  )  <, 

M,  is  positive  and  whose  intercept,  B,  is  negative,  and  also 
M^-\-4B  is  positive. 

XVIII.  y  =e°^(c  cosbx+dsmbx) ^ 61 

Values  of  x  form  an  arithmetical  series  and  the  points  repre- 
sented by  ( ^^\  ^^M  lie  on  a  straight  line  whose  slope, 
\  yk      yk  I 

M,  and  intercept,  B,  have  such  values  that  If 2+45 
is  negative. 


CONTENTS  7 


PAGE 


XIX.  y  =  ax'+bx^ 65 

Values  of  x  form  a  geometrical  series  and  the  points  repre- 
sented by  /-— \  ^^\  lie  on  a  straight  line  whose  slope, 

\  yt      yk  / 

M,    is    positive    and    intercept,   B,  negative,    and  also 
If  2 +45  positive. 

XlXa.  y  =  ax*c^ 72 

Values  of  x  form  a  geometrical  series  and  the  points  repre- 
sented by  Ixn,  log  ^^^\  lie  on  a  straight  line. 

CHAPTER  V 

XX.  y  =  aQ+aiCOSX-\-a2COS2x-\-a3  cos^x-\-  .  .  .  +ar  cos  rx 74 

-i-bismx+biSin  2x-\-b3sin  sx-h  .  .  .   -{-br  sin  rx 
Values  of  y  periodic. 

CHAPTER  VI 

Method  of  Least  Squares 90 

Apphcation  to  Linear  Observation  Equations. 
Application  to  Non-linear  Observation  Equations. 

CHAPTER  VII 

Interpolation mt, 100 

Differentiation  of  Tabulated  Functions. 

CHAPTER  VIII 

Numerical  Integration 114 

Areas. 
Volumes. 
Centroids. 
Moments  of  Inertia. 

APPENDIX 

Figures  I  to  XX 132-143 

Index 144 


EMPIRICAL   FORMULAS 


INTRODUCTION 

In  the  results  of  most  experiments  of  a  quantitative  nature, 
two  variables  occur,  such  as  the  relation  between  the  pressure 
and  the  volume  of  a  certain  quantity  of  gas,  or  the  relation 
between  the  elongation  of  a  wire  and  the  force  producing  it. 
On  plotting  the  sets  of  corresponding  values  it  is  found,  if  they 
really  depend  on  each  other,  that  the  points  so  located  lie 
approximately  on  a  smooth  curve. 

In  obtaining  a  mathematical  expression  which  shall  represent 
the  relation  between  the  variables  so  plotted  there  may  be  two 
distinct  objects  in  view,  one  being  to  determine  the  physical 
law  underlying  the  observed  quantities,  the  other  to  obtain 
a  simple  formula,  which  may  or  may  not  have  a  physical  basis, 
and  by  which  an  approximate  value  of  one  variable  may  be 
computed  from  a  given  value  of  the  other  variable. 

In  the  first  case  correctness  of  form  is  a  necessary  considera- 
tion. In  the  second  correctness  of  form  is  generally  considered 
subordinate  to  simplicity  and  convenience.  It  is  with  the 
latter  of  these  (Empirical  Formulas)  that  this  volume  is  mostly 
concerned. 

The  problem  of  determining  the  equation  to  be  used  is  really 
an  indeterminate  one;  for  it  is  clear  that  having  given  a  set  of 
corresponding  values  of  two  variables  a  number  of  equations 
can  be  found  which  will  represent  their  relation  approximately. 

Let  the  coordinates  of  the  points  in  Fig.  i  represent  different 
sets  of  corresponding  values  of  two  observed  quantities,  x  and  y. 
If  the  points  be  joined  by  segments  of  straight  lines  the  broken 

9 


10 


EMPIRICAL  FORMULAS 


line  thus  formed  will  represent  to  the  eye,  roughly,  the  relation 
between  the  quantities. 

It  is  reasonable  to  suppose,  however,  that  the  irregular  dis- 
tribution of  the  points  is  due  to  errors  in  the  observations,  and 
that  a  smooth  curve  drawn  to  conform  approximately  to  the 
distribution  of  the  points  will  more  nearly  represent  the  true 
relation  between  the  variables.  But  here  we  are  immediately 
confronted  with  a  difficulty.     Which  curve  shall  we  select?    a 


' 

u 

y'. 

/ 

^ 

^ 

y 

k 

^ 

Y 

^ 

^ 

/ 

/ 

a 

/ 

^ 

i 

C^ 

-r 

Fig.  I. 


or  J?  or  one  of  a  number  of  other  curves  which  might  be  drawn 
to  conform  quite  closely  to  the  distribution  of  the  points? 

In  determining  the  form  of  curve  to  be  used  reliance  must 
be  largely  placed  upon  intuition  and  upon  knowledge  of  the 
experiments  performed. 

The  problem  of  determining  a  simple  equation  which  will 
represent  as  nearly  as  possible  the  curve  selected  is  by  far  the 
more  difficult  one. 

Ordinarily  the  equation  to  be  used  will  be  derived  from  a 
consideration  of  the  data  without  the  intermediate  step  of 
drawing  the  curve. 

Unfortunately,  there  is  no  general  method  which  will  give 
the  best  form  of  equation  to  be  used.  There  are,  however,  a 
number  of  quite  simple  tests  which  may  be  applied  to  a  set  of 


INTRODUCTION  11 

data,  and  which  will  enable  us  to  make  a  fairly  good  choice  of 
equation. 

The  first  five  chapters  deal  with  the  application  of  these 
tests  and  the  evaluation  of  the  constants  entering  into  the 
equations.  Chapter  VI  is  devoted  to  the  evaluation  of  the 
constants  in  empirical  formulas  by  the  Method  of  Least  Squares. 
In  Chapter  VII  formulas  for  interpolation  are  developed  and 
their  applications  briefly  treated.  Chapter  VIII  is  devoted  to 
approximate  formulas  for  areas,  volumes,  centroids,  moments  of 
inertia,  and  a  number  of  examples  are  given  to  illustrate  their 
application. 

Figs.  I  to  XX  at  the  end  of  the  book  show  a  few  of  the  forms 
of  curves  represented  by  the  different  formulas. 

A  few  definitions  may  be  added. 

Arithmetical  Series.  A  series  of  numbers  each  of  which, 
after  the  first,  is  derived  from  the  preceding  by  the  addition 
of  a  constant  number  is  called  an  arithmetical  series.  The  con- 
stant number  is  called  the  common  difference 

6,        6.3,         6.6,  6.9,  7.2,        7-5  ••  . 

and 

18.0,        15.8,        13.6,        11.4,        9.2  .  .  . 

are  arithmetical  series.  In  the  first  the  common  difference  is 
.3,  and  in  the  second  the  common  difference  is  —  2.2. 

Geometrical  Series.  A  series  of  numbers  each  term  of  which, 
after  the  first,  is  derived  by  multiplying  the  preceding  by  some 
constant  multiplier  is  called  a  geometrical  series.  The  constant, 
multiplier  is  called  the  ratio. 

1.3,         2.6,        5.2,  10.4,        20.8,        41.6  .  .  . 

and 

100,         20,         4,  .8,  .16,  .032  .  .  . 

are  geometrical  series.  In  the  first  the  ratio  is  2,  and  in  the 
second  it  is  .2. 

Differences  are  frequently  employed  and  their  meaning  can 
best  be  brought  out  by  an  example. 


12 


EMPIRICAL  FORMULAS 


* 

y 

Ay 

A«y 

A»y 

A^ 

I 

10.2 

2 

11. 1 

0.9 
i.i 

0.2 

0.0 

3 

12.2 

0.2 

1.2 

4 
5 

l6.2 

1-3 

2.7 
1,8 

1.4 
-0.9 

1.2 
-2.3 

-35 
2.4 

6 

i8.o 

-0.8 

0.1 

2. 7 

I.O 

2.8 

7 

19,0 

2.0 

8 

22.0 

30 

In  the  table  corresponding  values  of  x  and  y  are  given  in 
the  first  two  columns.  In  the  third  column  are  given  the  values 
of  the  first  differences.  These  are  designated  by  A3'.  The  first 
value  in  the  third  column  is  obtained  by  subtracting  the  first 
value  of  y  from  the  second  value.  The  column  of  second  differ- 
ences, designated  by  A^y,  is  obtained  from  the  values  of  Ay  in 
the  same  way  that  the  column  of  first  differences  were  obtained 
from  the  values  of  y.  The  method  of  obtaining  the  higher  differ- 
ences is  evident. 


CHAPTER  I 

I.  y  =  a-^hx+cx^^-dx^-\-  .  .   .  +qx'' 

Values  of  x  form  an  arithmetical  series  and  A"y  constant. 

In  a  tensile  test  of  a  mild  steel  bar,  the  following  observa- 
tions were  made  (Low's  Applied  Mechanics,  p.  i88):  Diameter 
of  bar,  unloaded,  0.748  inch,  PF  =  load  in  tons,  rx:  =  elongation  in 
inches,  on  a  length  of  8  inches. 


w 

I 

' 

3 

4 

5 

6 

X 

Ax 

0.0014 
0.0013 

0.0027 
0.0013 

0.0040 
0.0015 

0.0055 
0.0013 

0.0068 
0.0014 

0.0082 

Plotting  W  and  x,  Fig.  2,  it  is  observed  that  the  points  lie 
very  nearly  on  a  straight  Hne.*  Indeed,  the  fit  is  so  good  that 
it  may  be  almost  concluded  that  there  exists  a  linear  relation, 
between  W  and  x.  From  the  figure  it  is  found  that  the  slope 
of  the  line  is  0.00137  and  that  it  passes  through  the  origin.  The 
relation  between  W  and  x  is  therefore  expressed  by  the  equation 

x=o.ooi:^'jW. 

The  observed  values  of  x  and  the  values  computed  by  the 
above  formula  are  given  in  the  table  below. 


w 

Observed  x 

Computed  x 

I 

2 

0.0014 
0.0027 

0.00137 
0.00274 

3 

4 
S 
6 

0.0040 
0.0055 
0.0068 
0.0082 

0.0041 I 
0 . 00548 
0.00685 
0.00822 

*  By  the  use  of  a  fine  thread   the  position  of  the  line  can  be  deter- 
mined quite  readily. 

13 


14 


EMPIRICAL  FORMULAS 


The  agreement  between  the  observed  and  the  computed 
values  is  seen  to  be  quite  good.  It  is  to  be  noted,  however,  that 
the  formula  can  not  be  used  for  computing  values  of  x  outside 


y 

y 

/ 

.0072 
0068 

/ 

/ 

/ 

(VMfl 

/ 

/ 

0056 

/ 

/ 

.0062 

A 

/ 

.0018 
.0014 

/ 

/ 

. 

/ 

003C 

/ 

y 

.0032 
.0028 
0021 

/ 

/ 

/ 

/ 

.0020 
.0012 

/ 

/ 

/ 

y 

/ 

/ 

.0001 

/ 

/ 

6   M) 


Fig.  2. 


the  elastic  limit.     In  the  experiment  6  tons  was  the  load  at  the 
elastic  limit. 

It  is  not  necessary  to  plot  the  points  to  determine  whether 
they  lie  approximately  on  a  straight  line  or  not.  Consider  the 
general  equation  of  the  straight  line 

y  =  mxArk' 

Starting  from  any  value  of  %,  give  to  x  an  increment,  A:*:,  and 
y  will  have  a  corresponding  increment.  Ay. 

y-\-t^y  =  m{x-\-l^x)-^'k\ 
y  —  mx-\-k\ 
Ay  =  wAa:. 


DETERMINATION   OF   CONSTANTS  15 

From  this  it  is  seen  that,  in  the  case  of  a  straight  line,  if  the 
increment  of  one  of  the  variables  is  constant,  the  increment  of 
the  other  will  also  be  constant. 

From  the  table  it  is  observed  that  the  successive  values  of 
W  differ  by  unity,  and  that  the  difference  between  the  successive 
values  of  x  is  very  nearly  constant.  Hence  the  relation  between 
the  variables  is  expressed  approximately  by 

x  =  mW+k, 

where  m  and  k  have  the  values  determined  graphically  from 
the  figure. 

By  the  nature  of  the  work  it  is  readily  seen  that  the  graphical 
determination  of  the  constants  will  be  only  approximate  under 
the  most  favorable  conditions,  and  should  be  employed  only 
when  the  degree  of  approximation  required  will  warrant  it. 
Satisfactory  results  can  be  obtained  only  by  exercising  great 
care.  Carelessness  in  a  few  details  will  often  render  the  results 
useless.  Understanding  how  a  graphical  process  is  to  be  carried 
out  is  essential  to  good  work;  but  not  less  important  is  the 
practice  in  applying  that  knowledge. 

In  experimental  results  involving  two  variables  the  values  of 
the  independent  variable  are  generally  given  in  an  arithmetical 
series.  Indeed,  it  is  seldom  that  results  in  any  other  form 
occur.  It  will  be  seen,  however,  that  in  many  cases  where 
the  values  of  the  independent  variable  are  given  in  an  arithmeti- 
cal series  it  will  be  convenient  to  select  these  values  in  a  geomet- 
rical series. 

As  a  special  case  consider  the  equation 

y  =  2—^x+x^. 

If  an  increment  be  assigned  to  x,  y  will  have  a  corresponding 
increment.  The  values  of  x  and  y  are  represented  in  the  table 
below.  Ay  stands  for  the  number  obtained  by  subtracting  any 
value  of  y  from  the  succeeding  value.     A^y  stands  for  the  num- 


16 


EMPIRICAL   FORMULAS 


ber  obtained  by  subtracting  any  value  of  A^^  from  its  succeeding 
value.     The  values  of  x  have  the  common  difference  0.5. 


* 

o.S 

I.O 

i-S 

2.0 

2.5 

30 

35 

4.0 

y 

0.7S 

0.00 

-0.2s 

0.00 

0.7S 

2.00 

3-75 

6.00 

Ay 

-0.7S 

-0.2s 

0.2s 

0.7S 

I -25 

I -75 

2.25 

A«y 

0.50 

0.50 

0.50 

0.50 

0.50 

0.50 

The  values  A^^;,  which  we  call  the  second  differences,  are 
constant. 

These  differences  could  equally .  well  have  been  computed 
as  follows: 

y-\-^y  =  2  —  3  (:r + ^x)  -\-{x-{-  Aa;)^, 
A)'=    —  3(Aic)  +  2x(Ax)  +  (Aa;)2, 

A3;+A2y=  -7,{^x)-\-2{x^-^x){^x)+{^xY, 

^y  =       2  (A:r)2  =  0.5  since  Aic  =  0.5. 

From  this  it  is  seen  that  whatever  the  value  of  Aic  (in 
y  =  2—2,x-\-x'^)  the  second  differences  of  the  values  of  y  are 
constant. 

Consider  now  the  general  case  where  the  wth  differences  are 
constant.  For  convenience  the  values  of  y  and  the  successive 
differences  will  be  arranged  in  columns.  The  notation  used  is 
self-explanatory. 


3^1 

^yi 

y2 

^^yi 

A>/2 

A3yi 

etc., 

3'3 

A2y2 

A4;yi 

Ays 

^^y2 

etc.. 

3'4 

A2y3 

^^y2 

Ay4 

^^yz 

etc., 

y-o 

A^'s 

A2>;4 

... 

y& 

DETERMINATION  OF  CONSTANTS  17 

From  the  above  it  is  clear  that 

y2=yi+Ayi, 

y3=y2+Ay2, 

=yi-\-Ayi+A(yi-\-Ayi), 
=yi-\-2Ayi-\-A^yi. 

y4:  =  y3+Ay3, 

=  3^1  +  2  A^^i  +  A^^'i  +  A(;yi  +  2A>;i + A^^^i), 
=yi+SAyi-\-sA^yi+A^yi. 

y5=y4.+Ay4., 

=  yi  +3A3/1  +sA^yi  -\-A^yi  +A(yi  +s^yi  +3^^yi  +A^yi) 
=  yi+4Ayi+6A^yi-i-4A^yi+A'^yi. 

In  the  above  equations  the  coefficients  follow  the  law  of 
the  binomial  theorem.  Assuming  that  the  law  holds  for  yt 
it  will  be  proved  that  it  holds  for  ^'i+i. 

By  hypothesis 

y^  =  yi  +  {k-i)Ay,  +  ^^~'^^^~^\^yi 

-^1 ^^ — 11 ^A3>;i+etc (i) 


If  this  equation  is  true,  then 

-2 

(k-l)(k-2)(k-3) 


2 


I A^yi+etc. 


+ A  [y,  +  (*  -  i)Aj', +  ^^-^i^^^A2yi 


i^^il^^pK^A3,,+etc.] 


yii-kAyi  +-i^ ^-A^y-\-^ r^ ^-A^y 

|2  13 


4 


18  EMPIRICAL  FORMULAS 

This  is  the  same  law  as  expressed  in  the  former  equation,  and 
therefore,  if  the  law  holds  for  y^,  it  must  also  hold  for  yt+i. 
But  we  have  shown  that  it  holds  for  y^,  and  therefore,  it  must 
hold  for  y5. 

Since  it  holds  for  y^  it  will  hold  for  yo.  By  this  process  it 
is  proved  that  the  law  holds  in  general. 

If  now  the  first  differences  are  constant  the  second  and 
higher  differences  will  be  zero,  and  from  (i) 

yk=yi-h(k-i)Ayi. 

If  the  second  differences  are  constant  the  third  and  higher 
differences  will  be  zero,  and  it  follows  from  (i)  that 

|2 

In  general,  then,  if  the  nth.  differences  are  constant 

yt=yi-^{k-i)Ayi-\-- P -A^yi+- ^^-i — - — ^A^yi 

+  .  .  .    \  (k-i){k-2)(k-s){k-4)  .  .  .   (k-n) 

The  law  requires  that  the  values  of  x  form  an  arithmetical  series, 
and  hence 

Xt=Xi-\-{k-i)Ax'j 
from  which  follows 

k='^^^^^+i .     (3) 

Ax  ^'^^ 

Substituting  this  value  of  k  in  equation  (2)  it  is  found  that 
the  right-hand  member  becomes  a  rational  integral  function  of 
Xic  of  the  wth  degree.     Equation  (2)  takes  the  form 

yt  =  a-^bXk-{-cx.j^-hdx^^-{-  .  .  .   -{-qxt*. 

Since  Xt  and  yt  are  any  two  corresponding  values  of  x  and  y 
the  subscripts  may  be  dropped  and  there  results  the  following 
law: 


DETERMINATION   OF   CONSTANTS 


19 


//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  nXh.  differences  of  the  cor- 
responding values  of  y  are  constant,  the  law  connecting  the  variables 
is  expressed  by  the  equation 


y  =  a-\-bx-\-cx'^-\-do(^-{- 


^qoiT. 


The  nth  differences  of  the  values  of  y  obtained  from  observa- 
tions are  seldom  if  ever  constant.  If,  however,  the  nth.  differ- 
ences approximate  to  a  constant  it  may  be  concluded  that  the 
relation  between  the  variables  is  fairly  well  represented  by  I. 

As  an  illustration  consider  the  data  given  on  page  131  of 
Merriman's  Method  of  Least  Squares.  The  table  gives  the 
velocities  of  water  in  the  Mississippi  River  at  different  depths 
for  the  point  of  observation  chosen,  the  total  depth  being  taken 
as  unity. 


Ah 


At  surface 
0.1  depth, 
0.2 

0.3  " 

0.4  " 

o.S  " 

0.6  " 

0.7  " 

0.8  " 

0.9  " 


1950 
2299 

2532 
2611 
2516 
2282 
1807 
1266 
0594 
9759 


+349 
+  233 
+  79 

-  95 
-234 
-475 
-541 

—  672 

-835 


-116 

-154 
-174 

-139 
-241 
-  66 
-131 
-163 


-  38 

-  20 

+  35 

-  102 

+  175 

-  65 

-  32 


+  18 
+  55 
-137 
+  277 
—  240 

+  Z3 


+  37 
—  192 

+414 
-517 
+  273 


From  the  above  table  it  is  seen  that  the  second  differences 
are  more  nearly  constant  than  any  of  the  other  series  of  differ- 
ences.   Of  equations  of  form  I, 

y  =  a-\-bx-\-cx^, 


where  x  stands  for  depth  and  y  for  velocity,  will  best  represent 
the  law  connecting  the  two  variables.  It  should  be  emphasized, 
however,  that  the  fact  that  the  second  differences  are  nearly 
constant  does  not  show  that  I  is  the  correct  form  of  equation 


20  EMPIRICAL  FORMULAS 

to  be  used.  It  only  shows  that  the  equation  selected  will 
represent  fairly  well  the  relation  between  the  two  variables. 

It  might  be  suggested  that  if  an  equation  of  form  I  with 
ten  constants  were  selected  these  constants  could  be  so  deter- 
mined that  the  ten  sets  of  values  given  in  the  table  would  satisfy 
the  equation.  To  determine  these  constants  we  would  sub- 
stitute in  turn  each  set  of  values  in  the  selected  equation  and 
from  the  ten  equations  thus  formed  compute  the  values  of  the 
constants.  But  we  would  have  no  assurance  that  the  equation 
so  formed  would  better  express  the  law  than  the  equation  of 
the  second  degree. 

For  the  purpose  of  determining  the  approximate  values  of 
the  constants  in  the  equation 

y  =  a+bx+cx'^ (i) 

from  the  data  given  proceed  in  the  following  way: 

Let  ii;=X+it;o, 

y=Y-\-yo, 

where  xo  and  yo  are  any  corresponding  values  of  x  and  y  taken 
from  the  data.     The  equation  becomes 

Y+yo  =  a+b{X+xo)+c(X+xoy 

=  a+bxo-\-  cxo^  -\-{b+ 2cxo)X-\-  cX^. 

Y={b+2cxo)X-{-cX^\ (2) 

since  yo  =  a-\-hxo-\-cX(i^.    Dividing  (2)  by  X  it  becomes 

—  ^b-\-2CXQ-\rcX (3) 

Y 

This  represents  a  straight  line  when  X  and  —  are  taken  as 

X 

coordinates.    The  slope  of  the  line  is  the  value  of  c  and  the 

intercept  the  value  of  b-\-2C0CQ.    The  numerical  work  is  shown 

in  the  table  and  the  points  represented  by  \X,  —\  are  seen 


DETERMINATION   OF   CONSTANTS 


21 


in  Fig.  3.     The  value  of  c  is  found  to  be  —0.76.     When  it:o  =  o, 
the  intercept,  0.44  is  the  value  of  h.     For  x=X,  the  value  of 


X 


.5 
Fig.  3. 


yo  is  taken  from  the  table  to  be  3.1950,  therefore  each  value  of 
Y  will  be  the  corresponding  value  of  y  diminished  by  3.1950. 


X 

y 

X 

Y 

Y 
X 

.44x-.i6x^ 

a=y-.44x 

Computed 
y 

.0 

3  1950 

0 

0.0000 

0 . 0000 

3.1950 

3.1948 

I 

3 

2299 

I 

0.0349 

0.3490 

0.0364 

3 

1935 

3.2312 

2 

3 

2532 

2 

0.0582 

0.2910 

0.0576 

3 

1956 

3.2524 

3 

3 

2611 

3 

0.0661 

0.2203 

0.0636 

3 

1975 

3.2584 

4 

3 

2516 

4 

0.0566 

O.1415 

0.0544 

3 

1972 

3.2492 

5 

3 

2282 

5 

0.0332 

0.0664 

0 . 0300 

3 

1982 

3.2248 

6 

3 

1807 

6 

-0.0143 

—  0.0238 

—  0 . 0096 

3 

1903 

3-1852 

7 

3 

1266 

7 

—  0.0684 

-0.0977 

—  0.0644 

3 

1910 

3.1304 

8 

3 

0594 

8 

-0.1356 

-0.1695 

-0.1344 

3 

1938 

3.0604 

9 

2.9759 

9 

—  0.  2191 

-0.2434 

—  0.2196 

3.1955 

2.9752 

10)31.9476 

a=  3.1948 

The  numbers  in  column  6  were  found  after  h  and  c  were  deter- 
mined in  Fig.  3.  The  sum  of  the  numbers  in  the  seventh  column 
divided  by  ten  gives  the  value  of  a.  In  the  last  column  are 
written  the  values  of  y  computed  from  the  formula 


3;  =  3.i948  +  .44X-76:v2 


(4) 


22  EMPIRICAL  FORMULAS 

TT  I    ^    I      <^      I    ^      I  Q 

Values  of  -  form  an  arithmetical  series  and  A^y  are  constant. 

Another  method  of  determining  the  constants  is  illustrated 
in  the  following  example:  Let  it  be  required  to  find  an  equation 
which  shall  express  approximately  the  relation  between  x  and  y 
having  given  the  corresponding  values  in  the  first  two  columns 
of  the  table  below. 


I 

2 

3 

4 

s 

6 

7 

8 

<y 

* 

y 

I  _ 

X 

X 

y 

Ay 

aV 

2 
"-■xi 

Com- 
puted y 

I.O 
1.2 

1-4 
1.6 
1.8 

2.0 
2.2 
2.4 

4.000 

2.889 
2.163 
1.656 
1.284 
1. 000 
0.777 
0.597 

1.0 

0.9 
0.8 

0.7 
0.6 

0.5 
0.4 
0.3 

1. 000 
I. HI 
1.250 
1.429 
1.667 
2.000 
2.500 

3-333 

4.00 
332 
2.68 
2.07 

I-5I 
1. 00 
0.52 
0.08 

-0.68 
—  0.64 
-0.61 
-0.56 
-0.51 
-0.48 
-0.44 

0.04 
0.03 
0.05 
0.05 
0.03 
0.04 

2.00 
1.50 
I.  14 
0.87 
0.67 
0.50 
0.36 
0.25 

4.000 
2.889 
2.163 
1.656 
1.284 
1. 000 
0.777 
0597 

In  column  3  are  given  values  of  -  in  arithmetical  series 

X 

and  the  corresponding  values  of  x  and  y  are  written  in  columns 
4  and  5  of  the  table.  The  values  of  y  were  read  from  Fig.  4. 
It  is  seen  that  the  second  differences  of  the  values  of  y  given 
in  column  7  are  nearly  constant,  and  therefore  the  relation 
between  the  variables  is  represented  approximately  by  the 
equation 

^-+0+<^^)-      •  •  •  •  (5) 

This  becomes  evident  if  x  be  replaced  by  -  in  I.    The  law 
may  then  be  stated : 

If  two  variables,  x  and  y,  are  so  related  that  when  values  of  - 

X 

are  taken  in  arithmetical  series  the  nXh  differences  of  the  corre- 


DETERMINATION   OF   CONSTANTS 


23 


sponding  values  oj  y  are  constant j  the  law  connecting  the  variables 
is  expressed  by  the  equation 


II 


s 

1 

^ 

5 

Values  of  -5 
.6 

r 
7 

8 

) 

1 

\ 

«} 

\ 

> 

y 

\ 

\ 

/ 

/ 

a 

\ 

\ 

/ 

o 

\ 

\ 

/ 

\ 

N 

/ 

^ 

^ 

V 

% 

/ 

I 

^ 

"\ 

V 

y^ 

•^ 

.^ 

/ 

< 

--^ 

^ 

/ 

_ 

_J 

1  1.2  1.4  L6  1.8  2.0  2. 

Values  of  x 


lA         2.6         2.8         3.0 


Fig.  4. 


If  in  equation  (5)  -  be  replaced  by  X,  then 


and 


y^^a^bX^cX^, 

By  subtracting  (6)  from  this  equation 

A>;  =  tAX+2c(AX)Z+c(AZ)2;     .     .     . 
and  from  (7) 

A3;+ A^^y  =  b^X  +  2c(AX)  (X+ AZ)  +c(AX)2.    . 

Subtracting  (7)  from  (8) 

A23;  =  2c(AX)2; 

A^y 
^     2(AZ)2- 


(6) 


.     (7) 
.     (8) 


24 


EMPIRICAL  FORMULAS 


From  column  7  it  is  seen  that  the  average  value  of  A^y  is 
0.04,  and  as  AJf  was  taken  — .  i , 


_  0.04 


Writing  the  equation  in  the  form 


■*■      =    T- 

^_ 

.iiiiiiii/" 

iiiiii=i=i 

J. — 

1 4- 

1 

.6 -I 

J. 

===i=i    III 

.3      A    .5     .6     .7     JJ     .9    1.0 


Fig.  5. 


it  is  seen  that  it  represents  a 
straight  line  when  -  andy— 

X 

2 

—  are  the  coordinates.     From 

x^ 

Fig.  5  &  is  found  to  be  3 
and  a  to  be  —  i.  The  for- 
mula is 


>'=-i+3(-)+2 


x2/' 


The  last  column  gives  the 
values  of  y  computed  from 
this  equation. 

The  following,  taken  from 
Saxelby's  Practical  Mathe- 
matics, page  134,  gives  the 
relation  between  the  poten- 
tial difference  V  and  the  cur- 
rent'-4  in  the  electric  arc. 
Length  of  arc  =2  mm.,  A  is 
given  in  amperes,  V  in  volts. 


A 

Observed  V. . 
1 

A 

Computed  V. 


1.96 
50-25 


.5102 
50.52 


2.46 
48.70 

.4065 

48.79 


2.97 
47.90 

•  3367 
47.62 


3-45 
47  50 

.2899 

46.84 


396 
46.80 

•2525 
46.22 


4.97 
45-70 


45  36 


5-97 
45.00 

•1675 

44.80 


6.97 
44.00 

•1435 
44  40 


7-97 
43.60 

-125s 
44.10 


9.00 
43-50 

.1111 
43-85 


DETERMINATION   OF   CONSTANTS 


25 


Fig.  6  shows  V  plotted  to  ^  as  abscissa.     The  slope  of  this 

line  is  12.5  divided  by  .75  on 6. 7.     The  intercept  on  the  V  —  axis 
is  42.     This  gives  for  the  relation  between  V  and  A 


F  =  42-| 


16.7 


Although  the 
points  in  Fig.  6  do 
not  follow  the  straight 
line  very  closely  the 
agreement  between 
the  observed  and  the 
computed  values  of  V 
is  fairly  good. 


r 

_J- 

__ 

^ 

B- 

. ■ 

"^ 

^ 

Fig.  6. 


III. 


-  =  a-\-bx-\-cx^-\-dx^-{-  .  .  .   -\-qx". 

y 


,nl 


Values  of  x  form  an  arithmetical  series  and  A'-  constant. 

y 

If  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  an  arithmetical  series  the  wth  differences  of  the  cor- 
responding values  of  -  are  constant,  the  law  connecting  the  variables 

y 

is  expressed  by  the  equation 


III  -  =  a^bx+cx'^+dx?+  .  .  .  +jx". 

y 

This  becomes  evident  by  replacing  ^^  in  I  by  -.  The  con- 
stants in  III  may  be  determined  in  the  same  way  as  they  were 
in  I. 


IV.     y^  =  a+bx+cx^-]-dx^+  .  .  .  -^-qx"". 
Values  of  x  form  an  arithmetical  series  and  A"  y"^  constant. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  an  arithmetical  series  the  nth.  differences  of  the  cor- 


26  EMPIRICAL  FORMULAS 

responding  values  of  y^  are  constanty  tlie  law  connecting  the  variables 
is  expressed  by  the  equation 

IV  'f  =  a-\-bx-{-coc^-\-dx^-\-  .  .  .  ^-qo^. 

This  also  becomes  evident  from  I  by  replacing  y  by  y"^. 

The  method  of  obtaining  the  values  of  the  constants  in 
formulas  III  and  IV  is  similar  to  that  employed  in  formulas  I 
and  II  and  needs  no  particular  discussion. 


CHAPTER  II 

V.     y  =  ab\ 
Values  of  x  form  an  arithmetical  series  and  the  values  of  y  a  geometrical 


senes. 


If  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  an  arithmetical  series  the  corresponding  values  of 
y  form  a  geometrical  series,  the  relation  between  the  variables  is 
expressed  by  the  equation 
V  y  =  ay. 

If  the  equation  be  written  in  the  form 
\ogy  =  \oga+([ogb)x, 

it  is  seen  at  once  that  if  the  values  of  x  form  an  arithmetical 
series  the  corresponding  values  of  log  y  will  also  form  an  arith- 
metical series,  and,  hence,  the  values  of  y  form  a  geometrical 
series. 

The  law  expressed  by  equation  V  has  been  called  the  com- 
pound interest  law.  If  a  represents  the  principal  invested,  b  the 
amount  of  one  dollar  for  one  year,  y  will  represent  the  amount 
at  the  end  of  x  years. 

The  following  example  is  an  illustration  under  formula  V. 

In  an  experiment  to  determine  the  coefficient  of  friction,  /z, 
for  a  belt  passing  round  a  pulley,  a  load  of  W  lb.  was  hung 
from  one  end  of  the  belt,  and  a  pull  of  P  lb.  appHed  to  the  other 
end  in  order  to  raise  the  weight  W.  The  table  below  gives  cor- 
responding values  of  a  and  //,  when  a  is  the  angle  of  contact 
between  the  belt  and  pulley  measured  in  radians. 


a 

IT 
2 

27r 
3 

6 

TT 

77r 
6 

47r 
3 

3^ 
2 

5^ 
3 

ii;r 
~6" 

P 

5.62 

6.93 

8.52 

10.50 

12.90 

15.96 

19.67 

24.24 

29.94 

27 


28 


EMPIRICAL  FORMULAS 


The  values  of  a  form  an  arithmetical  series  and  the  values 
of  P  form  very  nearly  a  geometrical  series,  the  ratio  being  1.23. 
The  law  connecting  the  variables  is 

The  constants  are  determined  graphically  by  first  writing 
the  equation  in  the  form 

log  P  =  log  a  -\-a  log  h 

and  plotting  the  values  of  a  and  P  on  semi-logarithmic  paper; 
or,  using  ordinary  cross-section  paper  and  plotting  the  values 
of  a  as  abscissas  and  the  values  of  log  P  as  ordinates.  Fig.  7 
gives  the  points  so  located.  The  straight  line  which  most 
nearly  passes  through  all  of  the  points  has  the  slope  .1733  and 
the  intercept  .4750.  The  slope  is  the  value  of  log  h  and  the 
intercept  the  value  of  log  a. 


2j0 

|1.5 
01.0 


Vi>r    %^  %Tr    TT      %7r   y^ir  y^ir    %ir  %ir 
Values  of  ct 

Fig.  7. 


log  a =0.4750, 
log  ^>  =  0.1733; 

b  =  i.4g. 


or 


The  formula  expressing  the  relation  between  the  variables  is 
P  =  3(i.49r, 

VI.     y  =  a+b(f. 

Values  of  x  form  an  arithmetical  series  and  the  values  of  Ay  form  a 
geometrical  series. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  first  differences  of  the  values 


DETERMINATION   OF   CONSTANTS 


29 


oj  y  form  a  geometrical  series,  the  relation  between  the  variables 
is  expressed  by  the  equation 

VI  y  =  a-Vbc\ 

By  the  conditions  stated  the  nth  value  of  x  will  be 

Xn  =  Xi-\-{n-i)  ^x, 

and  the  series  of  first  differences  of  the  values  of  y  will  be 

The  values  of  y  will  form  the  series 
yu       >'i+A>'i,       yl-\■^yl+r^yu 

yl-{-^yl^Vr^yl-\-r'^^yl-\-r^^yl-\-  .  .  .  -f-^""^A>'i. 
The  wth  value  of  y  will  be  represented  by 


A;yir^  .  .  .  I^yir 
yi-\-r^yi-\-r'^Lyi  . 


n-2 


yn=yi-\-^yi 
From  the  wth  value  of  x 

'  n  —  i  = 


i—r 

Xn       X\ 


Ax 

Substituting  this  value  in  the  above  equation  there  is  ob- 
tained 

Xn  -XI 


>'i+A3;i- 
a-\-bc', 


i—r 


where  a  stands  for  yi-\- 


Ayi 


b  for- 


Ayi 


r  Aa:  ^  and  c  for  r^^. 


Let  it  be  required  to  find  the  law  connecting  x  and  y  having 
given  the  corresponding  values  in  the  first  two  lines  of  the 
table. 


X 

o 

.  I 

.  2 

-3 

-4 

•5 

.6 

•  7 

■8 

-9 

I.O 

y 

Ay 

y 

1.300 

0. 140 
1.300 

1.440 
0.157 

1-439 

1-597 
0.177 

1-597 

1-774 
0.  200 

1-774 

1-974 
0.224 

1-973 

2.198 
0.254 
2.198 

2.452 
0.285 
2.452 

2-737 
0.323 
2.738 

3.060 
0.363 
3  -  059 

3-423 
0.407 
3.421 

3-830 
3-830 

30  EMPIRICAL  FORMULAS 

Since  the  values  of  A3;  form  very  nearly  a  geometrical  series 
the  relation  between  the  variables  is  expressed  approximately 
by 

y-a-^-bc". 

The  constants  in  this  formula  can  be  determined  graphically 
in  either  of  two  ways.  First  determine  a  and  then  subtract 
this  value  from  each  of  the  values  of  y  giving  a  new  relation 

y—a  =  bc'; 

which  may  be  written  in  the  logarithmic  form 

log  (y-a)  =log  b+x  log  c, 

and  b  and  c  determined  as  in  Fig.  7;   or,  determine  c  first  and 
plot  c*  as  abscissas  to  y  as  ordinate  giving  the  straight  line 

y  =  a+&(cO, 

whose  slope  is  b  and  whose  intercept  is  a. 

First  Method.  The  determination  of  a  is  very  simple. 
Select  three  points  P,  Q,  and  R  on  the  curve  drawn  through 
the  points  represented  by  the  data  such  that  their  abscissas 
form  an  arithmetical  series.     Fig.  8  shows  the  construction. 

P^{xo,a-^b(f^)', 

Q=(xo+Ax,  a-^bcfc""'); 

R=(xo-{-2Ax,  a+b(f'(^^). 
Select  also  two  more  points  5  and  T  such  that 

S^(xo-^Ax,a+b(f''); 

T={xo-\-2Ax,  a+b(f''c^''). 
The  equation  of  the  line  passing  through  Q  and  R  is 

y  = ^^ ^-x ^^ -{xQ-\-Ax)-^a-{-b(f'c^.     (i) 

Ax  Ax 


DETERMINATION    OF   CONSTANTS 


31 


The  equation  of  the  line  through  the  points  S  and  T  is 


y= 


Ax 


^x 


{x,^-^^x)+a+h(f\      .     (2) 


These  Hnes  intersect  in  a  point  whose  ordinate  is  a.  For, 
multiplying  equation  (2)  by  c^""  and  subtracting  the  resulting 
equation  from  (i)  gives 

y  =  a. 

y 
4. 


) 

y 

Y 

y 

y 

c 

^ 

T 

1 

a 

y^ 

^ 

^ 

^ 

p 

y' 

s 

^ 

\      d 

-^^ 

^ 

r^ 

^ 

0  .1  .2  .3         .4  .5  .6         .7  .8         .9        1.D 

Fig.  8. 

Fig.  8  gives  the  value  of  a  equal  to  0.2.     The  formula  now 
becomes 

log  (>'  — .2)  =log  h-\-x  logc. 

In  Fig.  9  log  {y  —  -2)  is  plotted  to  x  as  abscissa.     The  slcpe  of 
the  line  is  0.5185  which  is  the  value  of  log  c,  hence  c  is  equal  to 
2).^).     The  intercept  is  the  ordinate  of  the  first  point  or  0.0414, 
which  is  the  logarithm  of  h,  hence  h  is  equal  to  i.i. 
The  formula  is 

3'  =  o.24-i.i(3.3)^ 


32 


EMPIRICAL  FORMULAS 


or 


The  last  line  in  the  table  gives  the  values  of  y  computed  from 
this  formula. 

Second  Method.    For  any  point  {x,y)  the  relation  between 
X  and  y  is  expressed  by 

y^a+bc', 

and  for  any  other  point  (x+Ax,  y+Ay)  by 

y-\-Ay  =  a'-\-b(fc^. 

From  these  two  equations  is  obtained 

Ay^bcfic^-i) 

log  Ay = log  b{c^*  —  i)+x  log  c, 

6 

If  now  log  Ay  be 
plotted  to  X  as 
abscissa  a  straight 
line  is  obtained 
J  whose  slope  is  log  c. 
The  value  of  c  hav- 
ing been  determined, 
the  relation 

y=a+b(c') 

will  represent  a 
straight  line  pro- 
vided y  is  plotted 
to    (f    as    abscissa. 


.2     .3      .4      .5      .6      .7      .8 
Values  of  x 

Fig.  g. 


The  slope  of  this  line  is  b  and  its  intercept  a. 

VII.     \ogy  =  a+bc\ 

Values  of  x  form  an  arithmetical  series  and  the  values  of  A  log  y  form  a 
geometrical  series. 

If  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  first  differences  of  the  cor- 


DETERMINATION    OF   CONSTANTS  '33 

responding  values  of  log  y  form  a  geometrical  series,  the  relation 
between  the  variables  is  expressed  by  the  equation 

VII  log  y  =  a+bc\ 

This  is  at  once  evident  from  VI  when  y  is  replaced  by  log  y. 
The  only  difference  in  the  proof  is  that  instead  of  the  series 
of  differences  of  y  the  series  of  differences  of  log  y  is  taken. 

VIII.    y  =  a+bx+cd\ 

Values  of  x  form  an  arithmetical  series  and  the  values  of  A^y  form  a 
geometrical  series. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  values  of  the  second  differ- 
ences of  the  corresponding  values  of  y  form  a  geometrical  series, 
the  relation  between  the  variables  is  expressed  by  the  equation 

VIII  y  =  a+bx+cd\ 
The  wth  value  of  x  is  represented  by 

Xn=Xi  +  {n—l)^X. 

The  values  of  y  and  the  first  and  second  differences  may  be 


in  columns 

yi 

^yi 

y2 

Ay2 

A2yi 

3^3 

Ays 

A2y2 

3^4 

Ay4 

A2y3 

y^ 

Ays 

A2y4 

y^ 

etc. 

etc. 

etc. 

34  EMPIRICAL  FORMULAS 

Since  the  second  differences  of  y  are  to  form  a  geometrical 
series  they  may  be  written 

A2^i,         rt^yi,         r^t^yi,         r^i^yx  .  .  .  f'-^t^yx. 

The  series  of  first  differences  will  then  be 

Ayi,     Ly\  H-A^^/i,      Ly^  -^r^^yi  ■\-r^^yi,     Aji  ^-L^yi  +rA2;yi  -^r^b^y^ 

Lyi^L^yx-^-rt^yx^-r^l^yx^-  .  .  .   -^f'-^l^yy^. 

The  «th  value  of  y  will  be  equal  to  the  first  value  plus  all 
the  first  differences.  For  convenience  the  «th  value  of  y  is 
written  in  the  table  below. 

%=^y\ 
+A>'i 

■\■^yl-Vl^yl 
-{■t^yx  •\-i^y\  ■\-rl^y\ 
-\-Ly\-\-l^y\-\-ri^y\-\-r^^yx 
+ A^'i  +  I^yx + rt^yi  +  r^^yi  +  r^t^yx 


-\-t^y\-\-!^yi-\-r!^y\-\-r^^y\-\-r^t^y\-\-  .  .  .  ■\-r''~^l^yi. 
Adding  gives 

yn  =  yi  +  {n-i)Ayi+A^yi\ 1 1 \- 

11— r     i—r      i—r      i—r 

i—r^  I— r""^! 

i—r  i—r  J 

The  first  two  terms  on  the  right-hand  side  represent  the  sum 
of  all  the  terms  in  the  first  column  of  the  value  of  yn.  The 
remaining  terms  contain  the  common  factor  L^yi,  The  terms 
inside  the  bracket  are  easily  obtained  when  it  is  remembered 
that  each  line,  omitting  the  first  term,  in  the  value  of  y  form  a 
geometrical  series.  It  is  easily  seen  that  the  value  of  y„  may  be 
written 


DETERMINATION   OF   CONSTANTS 


35 


yn=^yi+(n'-i)Ayi+^^{n-2)-^^^^(r+r^+r^+  .  .  . +r"-') 


i—r 


i—r 


=yi  +  (n-i)Ayi 


A^y^ 


-{n-i) ^ 

I—r  I—r     I—r 


=.4+5(w-i)+Cr"-'; 

where 

A^y 


A=y,--^^yL-,  B  =  Ay,+^,  and  C  = 


A2y, 


A^yi 


(i-r) 


I— r 


(i-ry 


From  the  value  of  Xn  is  obtained 

Xn—Xl 


n—1  =- 


Ax 


Substituting  this  in  the  value  of  yn  it  is  found 

y„=A-^B^^^^^^+Cr  ^' 
Ax 

=  a+bXn-{-cd''''. 

Since  x„  and  yn  stand  for  any  set  of  corresponding  values 
of  :*:  and  y  the  resulting  formula  is 

VIII  y  =  a+bx-\-cd\ 

In  the  first  two  columns  of  the  following  table  are  given 
corresponding  values  of  x  and  y  from  which  it  is  required  to 
find  a  formula  representing  the  law  connecting  them. 


* 

y 

Ay 

A'y 

log  Ah 

(2.00)*   y— I. 

01(2.00)^ 

Computed  y 

.0 

1.500 

048 

023 

-1.6383 

1. 000 

490 

1.492 

.2 

1.548 

071 

026 

—  I 

5850 

1. 149 

388 

550 

•  4 

1.619 

097 

028 

-I 

5528 

1.320 

286 

620 

.6 

1.716 

125 

034 

-I 

4685 

1. 517 

184 

715 

.8 

1. 841 

159 

039 

-I 

4089 

1.742 

082 

841 

I.O 

2.000 

198 

043 

-I 

3665 

2.000     — 

020 

999 

1.2 

2.198 

241 

051 

-I 

2924 

2.300     — 

125 

2 

196 

1.4 

2.439 

292 

059 

—  I 

2291 

2 . 640    — 

227 

2 

440 

1.6 

2.731 

351 

067 

—  I 

1739 

3.032 

331 

2 

735 

1.8 

3.082 

418 

3.482 
4.000    — 

435 
540 

3 

085 

2.0 

3.500 

3 

506 

36 


EMPIRICAL  FORMULAS 


Since  the  values  of  x  form  an  arithmetical  series  and  the 
second  differences  of  the  values  of  y  form  approximately  a 
geometrical  series,  it  is  evident  that  the  relation  between  the 
variables  is  fairly  well  represented  by 

y  =  a-\-hx-\-cd\ 
Taking  the  second  difference 

log  A2y  =  log  c(^^  - 1)2+ (log  d)x. 


or 


Plotting  the  logarithms  of  the  second  differences  of  y  from 
the  table  to  the  values  of  x,  Fig.  lo,  it  is  found  that  log  d  =  .3000 


r- 

^ 

^ 

^ 

H 

^ 

^ 

y 

^' 

■  § 

.4T 

00 

X 

X 

^ 

"^ 

X 

J 

M 

X 

.1 

^'•' 
*l.l 

y 

^ 

X 

"g. 

?15 

^ 

\y 

X 

A 

A 

hN 

"^^ 

^ 

'y' 

\ 

y 

^ 

0       ^      ^       ,6       .8      1.0     1.2     1^     1.6     1.8     2jO 
Valuea  of  <o 

Fig.  10. 


or   J  =  1.995,    approximately    2.     The   intercept   of   this   line, 

— 1.6500,  is  equal  to  log  c{d^  —  iY. 

Since 

J=2, 
.02239  =  ^(2-^-1)2, 
C  =  I.OII. 


DETERMINATION   OF   CONSTANTS  37 

Plotting  3;— (1.01)2*  to  Xy  Fig.  lo,  the  values  of  a  and  h 

are  found  to  be 

a=     0.5, 

^>= -0.515. 

The  formula  derived  from  the  data  is 

3;  =  o.5-o.5i5a;+(i.oi)2^ 

In  the  last  column  of  the  table  the  values  of  y  computed 
from  the  formula  are  written  down.  Comparing  these  values 
with  the  given  values  of  y  it  is  seen  that  the  formula  reproduces 
the  values  of  ^^  to  a  fair  approximation. 

IX.    3;  =  10'^+'^+^^'. 
Values  of  x  form  an  arithmetical  series  and  A  2  log  y  constant. 

If  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  second  differences  of  the 
values  of  log  y  are  constant,  the  relation  between  the  variables  is 
expressed  by  the  equation 

IX  y  =  id'-^''^^''\ 

This  becomes  evident  from  I  when  y  is  replaced  by  log  y. 

\o%y  =  a-\-bx-\-cx^, 

which  represents  a  parabola  when  log^*  is  plotted  to  x.  The 
constants  are  determined  in  the  same  way  as  they  were  in 
formula  I. 

X.    y  =  ks'g^. 
Values  of  -x  form  an  arithmetical  series  and  values  of  A^  log  y  form  a 
geometrical  series. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  second  differences  of  the 
corresponding  values  of  log  y  form  a  geometrical  series,  the  relation 
between  the  variables  is  expressed  by  the  equation 


38  EMPIRICAL  FORMULAS 

This  becomes  evident  by  taking  the  logarithms  of  both  sides 
and  comparing  the  equations  thus  obtained  with  VIII.  X 
becomes 

log  >'  =  log  ^+(log  5)a;4-(log  g)d\ 

This  is  the  same  as  VIII  when  y  is  replaced  by  log  y,  a  by 
log  k,  b  by  log  5,  and  c  by  log  g* 

XI.    y  = 


a-^bx-j-cx^ 
Values  of  x  form  an  arithmetical  series  and  A'-  are  constant. 


//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are  taken  in  an  arithmetical  series  the  second  differences  of  the 

X 

corresponding  values  of  -  are  constant,  the  relation  between  the 

y 

variables  is  expressed  by  the  equation 


XI 


a+bx-\-cx^ 
Clearing  equation  XI  of  fractions  and  dividing  by  y 

-==a-\-bx-\-cx^. 

y 

X 

This  is  of  the  same  form  as  I,  and  when  -  is  replaced  by  y 

y 

the  law  stated  above  becomes  evident. 
If  a  is  zero  XI  becomes 

_     I 

^~b^x 

which,  by  clearing  of  fractions  and  dividing  by  y,  reduces  to 

-  =  b-{-cx, 

y 

2l  special  case  of  III. 

*  For  an  extended  discussion  of  X  see  Chapter  VI  of  the  Institute  of 
Actuaries'  Text  Book  by  George  King. 


DETERMINATION   OF   CONSTANTS 


39 


If  c  is  zero  XI  becomes  a  special  case  of  XVI,  or 
-  =  a-]rhx, 

y 

which  is  a  straight  line  when  -  is  plotted  to  x. 

y 

Corresponding  values  of  x  and  y  are  given  in  the  table  below, 
find  a  formula  which  will  express  approximately  the  relation 
between  them. 


X 

y 

X 

y 

y 

A^.           X 

y 

Y 

Y           X 
X         ^ 

-2.SX^ 

Com- 
puted y 

o 

o.ooo 



... 



0.000 

.1 

1-333 

0.075 

100 

.050   — 

9 

-  2 . 703 

3 

003 

050 

1.329 

2 

1 .143 

0^175 

150 

.050   — 

8 

-2.603 

3 

254 

075 

1. 140 

3 

0.923 

0.325 

200 

.050   — 

7 

-2.453 

3 

504 

100 

0.929 

4 

0.762 

0.52s 

250 

-050   — 

6 

-2.253 

3 

755 

125 

0.760 

5 

0.645 

0.775 

300 

-051   - 

5 

-2.003 

4 

006 

150 

0.644 

6 

0.558 

I -075 

351 

-049   - 

4 

- 1 . 703 

4 

257 

175 

0.558 

7 

0.491 

1.426 

400 

.047   - 

3 

-1-352 

4 

507 

201 

0.491 

8 

0.438 

1.826 

447 

-058   — 

2 

-0.952 

4 

760 

226 

0.438 

9 

0.396 

2.273 

50s 

.040   — 

I 

-0-503 

5 

030 

248 

0.395 

o 

0.360 

2.778 

545 

-054 

0 

0.000 

0.360 

I 

0.331 

3323 

599 

.056 

I 

0.545 

5 

450 

298 

0.331 

2 

0.306 

3-922 

655 

-051 

2 

1. 144 

5 

720 

332 

0.30s 

3 

0.284 

4-577 

706 

-035 

3 

1.799 

5 

997 

352 

0.284 

4 

0.265 

5-283 

741 

4 

2.505 

6 

262 

383 

0.265 

1-5 

0.249 

6.024 



5 

3.246 

6 

492 

399 

0.249 

The  values  of  x  form  an  arithmetical  series  and  since  the 

X 

second  differences  of  -  are  nearly  constant  the  values  of  y  will 

y 

be  fairly  well  represented  by 


y= 


a-\-bx-\-cx^^ 


or 


=  a-\-hx+cx'^. 


This  represents  a  parabola  when  -  is  plotted  to  x. 

y 

Let  X  =  x—i, 


F  =  --2.778. 

y 


40  EMPIRICAL  FORMULAS 

From  these  equations  are  obtained 

-  =  F+2.778. 

y 

The  formula  becomes 

Since  the  new  origin  lies  on  the  curve 

a-\-b-\-c  =  2.^^2>, 


a     ^     ^     A     .5     .6     .7     A     J    LO  U   U  L3  JLi  1.5    « 


Fig.  II. 
the  equation  reduces  to 

Y  =  {h+2c)X+cX^, 

Y 


or 


X 


=  b+2c+cX. 


Y  . 


This  represents  a  straight  line  when  —  is  plotted  to  X.    The 

value  obtained  for  c  from  P'ig.  ii  is  2.5.     The  value  of  h  could 
be  obtained  from  the  intercept  of  this  line  but  the  approximation 


DETERMINATION   OF   CONSTANTS  41 


will  be  better  by  plotting  —  2.5ii;2  to  x.    In  this  way  is  obtained 
the  line 

—  2.<x^==a-\-bx. 

y 

From  the  lower  part  of  Fig.  ii  the  values  of  a  and  b  are 
found  to  be 

a  =  .o2S, 

^  =  •2525. 

Substituting  the  values  of  the  constants  in  XI  the  formula 
becomes 

X 


y= 


.025 +  .25250: -1-2. 5x2* 


In  the  last  column  of  the  table  the  values  of  y  computed 
from  this  equation  are  given  and  are  seen  to  agree  very  well 
with  the  given  values. 


^\c  ^ 


/ 


CHAPTER  III 

XII.    y=^a^. 

Values  of  x  form  a  geometrical  series  and  the  values  of  y  form  a 
geometrical  series. 

//  two  variables,  x  and  y,  are  so  related  that  when  the  values  of 
X  are  taken  in  a  geometrical  series  the  corresponding  values  of  y 
also  form  a  geometrical  series,  the  relation  between  the  variables  is 
expressed  by  the  equation 

XII  y=^ax\ 

From  the  conditions  stated  equations  (a)  and  ib)  are  obtained. 

:r„=xir"-^ (a) 

yn=yiR''-\ {b) 

where   r  is   the  ratio  of  any  value  of  x  to  the  preceding  one 
and  R  is  the  ratio  of  any  value  of  y  to  the  preceding  one. 
Taking  the  logarithm  of  each  member  of  {a) 

\ogXn  =  \ogxi  +  {n-i)\ogr, 

j^     ^._log:rn-logXi 
logr 

Also  by  substituting  this  value  of  n  —  i  in  the  value  of  yn  in 
equation  {b), 

log  xn  -log  XI 

yn=yiR     '°^'" 

log  XI  I       1    \  log  Xn 

=yiR    ^°«'-  U'°«7 


=  aid 


log  X 


DETERMINATION   OF   CONSTANTS 


43 


where 

and 

log  x« 

IO^=i2logr^ 

The  following  data  (Bach,  Elastizitat  und  Festigkeit)  refer 
to  a  hollow  cast-iron  tube  subject  to  a  tensile  stress;  x  represents 
the  stress  in  kilogrammes  per  square  centimeter  of  cross-section 
and  y  the  elongation  in  terms  of  -g^o  cm.  as  unit. 


X 

9-79 

20.02 

40.47 

60.92 

81.37 

101.82 

204 . 00 

408.57 

y 

0-33 

0.695 

I  530 

2.410 

3-295 

4.185 

8.960 

19.490 

log  X. . . 

o . 9908 

I. 3014 

1.6072 

I . 7847 

I. 9104 

2.0078 

2.3096 

2.6113 

log  y. .  . 

—0.4815 

-0.1580 

0.1847 

0.3820 

0.5178 

0.6217 

0.9523 

I . 2898 

Comp. 

y.... 

0.324 

0.714 

I -541 

2.416 

3323 

4.252 

9.132 

19 . 600 

Selecting  the  values  of  x  which  form  a  geometrical  series, 
or  nearly  so,  it  is  seen  that  the  corresponding  values  of  y  form 
approximately  a  geometrical  series,  and,  therefore,  the  relation 
between  the  variables  is  expressed  by  the  equation 


or 


y  =  ax  , 
log  3;  =  log  a+b  \ogx. 


If  now  logy  be  plotted  to  logo:  the  value  of  b  will  be  the 
slope  of  the  line  and  the  intercept  will  be  the  value  log  a.  Fig. 
12  gives  &  =  I.I.  In  computing  the  slope  it  must  be  remembered 
that  the  horizontal  unit  is  twice  as  long  as  the  vertical  unit. 
The  intercept  is  —1.5800  or  8.4200  —  10,  which  is  equal  to 
log  0.0263.     The  formula  is 

y  =  .0263:x:'*\ 

The  values  of  y  computed  from  this  equation  are  written 
in  the  last  line  of  the  table.  They  agree  quite  well  with  the 
observed  values. 


44 


EMPIRICAL  FORMULAS 


XIII.    y-a-\-b\ogx-^c\og^x 
Values  of  log  x  form  an  arithmetical  series  and  A'y  constant. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of 
log  X  are  taken  in  an  arithmetical  series  the  second  differences  of 
the  corresponding  values  of  y  are  constant  the  relation  between  the 
variables  is  expressed  by  the  equation 


XIII 


y  =  a-\-b  log  x-\-c  \o^x. 


This  becomes  evident  from  I  by  replacing  x  by  log  x.    The 
law  can  also  be  stated  as  follows:  If  the  values  of  x  form  a  geo- 


1.0 


o 
|.2 

5  i> 


y 

,^ 

y 

_^ 

y^ 

X- 

^ 

^ 

x' 

^ 

i^ 

,^ 

^ 

^^ 

^ 

^ 

y^" 

y 

/ 

^ 

.8    ^9      1     1.1   1.2  1.3  1.4   1.5  1.6   1.7    1,8  1.9     2     2.1  2.2    2.3   2.4   2.5  2.6  2.7   2.8 
Values  of  log  a: 

Fig.  12. 

metrical  series  and  the  second  differences  of  the  corresponding 
values  of  y  are  constant  the  relation  between  the  variables  is 
expressed  by  the  equation 

y  =  a-\-b\ogx-\-c\o^x. 
If  c  is  zero  the  formula  becomes 

y=^a-\-b\Qgx, 
which  is  V  with  x  and  y  interchanged. 


DETERMINATION   OF   CONSTANTS  45 

Formula  XIII  represents  a  parabola  when  y  is  plotted  to 
logic.  The  constants  are  determined  in  the  same  way  as  the 
constants  in  I. 

XIV.    y  =  a^hx\. 

Values  of  x  form  a  geometrical  series  and  values  of  Ay  form  a  geometrical 
series. 

Ij  two  variables,  x  and  y,  are  so  related  that  when  the  values  of 
X  are  taken  in  a  geometrical  series  the  first  differences  of  the  cor- 
responding values  of  y  form  a  geometrical  series,  the  relation  between 
the  variables  is  expressed  by  the  equation 

XIV  y  =  a+bx\ 

As  in  XII  the  n\h  value  of  x  is 

Xn=xir''~'^ {c) 

The  series  of  first  differences  of  y  may  be  written 

A3/1,        ^yiR,        AyiR^y        AyiR^  .  .  .  AyiR"-^, 

and  the  values  of  y  are 

yi,     yi+Ayi,    yi+Ayi+AyiR,    yi+Ayi+AyiR+AyiR^  .  .  . 

yi-\-Ayi+AyiR+AyiR^+AyiR^+  .  .  .  -^-AyiR^'-K 

That  is  the  nth.  value  of  y  will  be 

yn  =  yi+Ayi+AyiR+AyiR^-\-AyiR3-{-  .  .  .   +AyiR''-'' 
=yi-\-Ayi{i+R+R^+R^+  .  .  .  -{-R-^) 

=  yi-\-Ayi- — - (d) 

I  — K 

Taking  the  logarithm  of  each  member  of  (c), 
loga;„  =  loga:i  +  (w-i)  logr 

log  r 


46 


EMPIRICAL   FORMULAS 


Substituting  this  value  of  »— i  in  the  wth  value  of  y  given 
in  id), 


yn=^yi-{-^yi 


log  xn  -log  Ji 

1-R     '"«'• 


i-R 


log  XI  /        1    \  log  Xn 


.  .  log  Xl  /  1     \ 


=a+6io^"«"''' 

Let  it  be  required  to  find  the  law  connecting  x  and  y  having 
given  the  values  in  the  first  two  lines  of  the  table. 


X 

2 

3 

4 

5 

6 

7 

8 

y 

4.21 

5.2s 

6.40 

7.65 

8.96 

10.36 

ii.8i 

log* 

.3010 

.4771 

.6021 

.6990 

.7782 

•  8451 

.9031 

X 

2 

2.5 

3125 

3  906 

4.883 

6. 104 

7.630 

y 

4.210 

4.720 

5.388 

6.290 

7.515 

9.110 

11.275 

log* 

.3010 

.3979 

.4948 

.5918 

.6887 

.7856 

^y 

•  Sio 

.668 

.902 

I.  225 

1-595 

2.165 

.... 

log  Ay 

-  .2924 

-.1752 

-.0448 

.0881 

.2028 

•3358 

.... 

y-2.72 

1.49 

2.53 

3.68 

4.93 

6.24 

7.64 

9.09 

log(y-2.72) 

.1732 

•  4031 

.5658 

.6928 

.7952 

.8831 

.9586 

Computed  y 

4.21 

S-25 

6.41 

7.65 

8.98 

10.36 

II. 81 

In  the  fourth  Hne  values  of  x  are  given  in  a  geometrical 
series  with  the  ratio  1.25.  In  the  fifth  line  are  given  the  cor- 
responding values  of  y  read  from  Fig.  13.  The  first  differences 
of  the  values  of  y  are  written  in  the  seventh  line.  These  differ- 
ences form  very  nearly  a  geometrical  series  with  the  ratio  1.336. 
Since  the  ratio  is  nearly  constant  the  law  connecting  x  and  y 
is  fairly  well  represented  by  the  equation 

y  =  a-{-hx'^. 

There  are  two  methods  which  may  be  employed  for  deter- 
mining the  values  of  the  constants,  either  one  of  which  may 
serve  as  a  check  on  the  other. 


DETERMINATION   OF   CONSTANTS 


47 


First  Method.  Select  three  points,  A,  P,  and  Q  on  the 
curve,  Fig.  13,  such  that  their  abscissas  form  a  geometrical 
series  and  two  other 
points,  R  and  S,  such 
that  R  has  the  same 
ordinate  as  A  and  the 
same  abscissa  as  P,  5 
the  same  ordinate  as  P 
and  the  same  abscissa 
as  Q.  The  points  may 
be  represented  as  fol- 
lows: 

^  =  (^0,  a-{-hxo)\ 
P={xor,  a-{-bxoY); 
Q^ixor^a+bxo'r^'): 
R={xor,  a-\-bxo'); 
S^(xor^a-\-bxoY). 


The  equation  of  the 
line  passing  through  P 
and  Q  is 


12 

A 

// 

V 

9 

// 

/ 

/ 

0 

/ 

r 

p7 
> 
G 

5 

4 

} 

/ 

^ 

/ 

/ 

^ 

^ 

A, 

v 

^ 

^pj 

x^ 

/      1 

X 

A 

3 

V 

I 
alues  ol 

c 

X 

' 

8 

y= 7^ -^x-{-a  — 


Fig.  13. 
bxoY{r'-r) 


xor{r—i)  r—i 

The   equation   of   the  line  passing  through  the  points  R  and 
5  is 

^_bxo\r'-i)^^^     bxQ^r'-r) 


Xor(r-i) 


r  —  i 


These  two  lines  intersect  in  a  point  whose  ordinate  is  a.  In 
Fig.  13  xo  is  taken  equal  to  2  and  r  equal  to  2.  The  value  of 
a  is  found  to  be  2.72.     The  formula  then  becomes 


or 


y  —  2.'j2  =  bx'', 
log  ()'  — 2.72)=log  b-{-clogx. 


48 


EMPIRICAL  FORMULAS 


In  Fig.  14  log  (>>-- 2.72)  is  plotted  to  log  x  and  b  and  c 
determined  as  in  XII.  It  is  seen  that  the  points  lie  very  nearly 
on  a  straight  line.     The  values  of  c  and  h  are  read  from  Fig.  14. 

c=i.3; 

log  6  =  9.7840— 10; 

h=.6i. 

The  law,  connecting  x  and  y  then  is 

y  =  2.']2-\-.6ix^-^. 


.5 
.4 

.2 
*»  1 
lo 

K 

i3 
-A 
-16 

-.6 


/         / 

J^      / 

-y      ^^ 

z      z 

7       / 

jl       V          V 

i^     V 

±  z    ,z 

/         7 

7          J^ 

/           / 

z     z 

7          / 

^    ^^ 

.4      .5      .6       .7 
Values  of  log  x 

Fig.  14. 


.9    1.0 


.6 


.4  <n 

a> 

.2 
.1 
0 


The  values  of  y  computed  from  this  formula  are  written  in 
the  last  line  of  the  table. 

Second  Method.     From  the  equation 


y  =  a-\-hx''- 


DETERMINATION   OF  CONSTANTS  49 

we  have 

y-{-Ay  =  a-\-bx'^r^; 

log  Ay  =  log  h{r''-i)  +c  log  x. 

This  is  the  equation  of  a  straight  line  when  log  Ay  is  plotted 
to  \ogx.  Fig.  14  shows  the  points  so  plotted  and  from  the 
line  drawn  through  them  the  values  of  h  and  c  are  obtained. 

c  =  i.3, 

a  is  found  by  taking  the  average  of  all  the  values  obtained 

from  the  equation 

a=y  —  .6i^-^. 

a  is  equal  to  2.72. 

XV.    >'  =  aio*''. 

Values  of  X  form  a  geometrical  series  and  A  log  y  form  a  geometrical 
series. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  a  geometrical  series  the  first  differences  of  the  cor- 
responding values  of  log  y  form  a  geometrical  series,  the  relation 
between  the  variables  is  expressed  by  the  equation 

XV  y  =  aid>^\ 

This  equation  written  in  the  logarithmic  form  is 

log  y  =  \og  a-{-bxf . 

Comparing  this  with  XIV  it  is  evident  that  if  the  values  of  x 
form  a  geometrical  series  the  first  differences  of  the  corre- 
sponding values  of  log  y  also  form  a  geometrical  series. 

In  an  experiment  to  determine  the  upward  pressure  of  water 
seeping  through  sand  a  tank  in  the  form  shown  in  Fig.  15  was 
filled  with  sand  of  a  given  porosity  and  a  constant  head  of 


50 


EMPIRICAL   FORMULAS 


water  of  four  feet  maintained.*    The  water  was  allowed  to  flow 
freely  from  the  tank  at  A.    The  height  of  the  column  of  water 

in  each  glass  tube, 
six  inches  apart, 
was  measured.  In 
the  table  below  x 
represents  the  dis- 
tance of  the  tube 
from  the  water  head 
In  feet,  and  y  the 
height  of  the  column 
of  water  in  the  tube, 
also  in  feet.  It  is 
required  to  find  the 
law  connecting  x 
and  y. . 


Fig.  15. 


Tube 

I 

3 

3 

4 

S 

6 

7 

8 

9 

X 

0 

•  5 

I.O 

i.S 

2.0 

2.5 

3.0 

3-5 

4.0 

.  y 

^■H  , 

2.30 

2.20 

2.00 

1.66 

1.24 

0.84 

0.54 

0.28 

logy 

.3636 

•  3617 

.3424 

.3010 

.2201 

.0934 

-    .0757 

-    .2676 

-   .5528 

X 

•  S 

1.0 

2.0 

4.0 

y 

2.30 

2.20 

1.66 

.28 

logy 

.3617 

•  3424 

.2201 

-   .5528 

A  logy 

-   .0193 

-   .1223 

-   .7729 

logx 

—   .3010 

.0000 

.3010 

log  ( -A  logy) 

-1.7144 

-   .9126 

—   .1119 

,      **"._« 

.0000 

—   .0036 

—   .0228 

—   .0673 

-   .1449 

—   .2627 

-    .4272 

-    .6445 

-   .9201 

log  (y-6x«) 

.3b3b 

■  3653 

•  3652 

.3683 

•  3650 

.3561 

.3515 

.3769 

.3673 

Computed  y 

2.314 

2.29s 

2.195 

1.982 

1.658 

1.264 

.865 

.525 

.278 

In  the  fifth  line  values  of  x  are  selected  in  a  geometrical 
series  and  the  corresponding  values  of  y  written  in  the  next 
line.  In  Fig.  16  log  (—A  log  y)  is  plotted  to  logrr.  The 
points  lie  on  a  straight  line.  On  account  of  the  small  number 
of  points  used  in  the  test  we  select  formula  XV  on  trial. 
From  the  formula 


it  follows  that 


y  =  aior 


log  y  =  \og  a+bx^ 


*  Coleman's  Thesis,  University  of  Michigan. 


DETERMINATION   OF   CONSTANTS  51 

logyk^\oga-\-bxt 
log  yk+ 1  =  log  a + bxtY 
A\ogyk  =  bxt!'(r'-i) 
log  (A  log  y)  =log  b{r''-i)-\-c  log  x. 

If  A  log  y  is  negative  b  is  negative,  in  which  case  it  is  only- 
necessary  to  divide  the  equation  by  —  i  before  taking  the 
logarithms  of  the  two  members  of  the  equation. 


-1  0  .1 

Values  of'log  x 


Fig.  I 6. 


The  last  equation  above  represents  a  straight  line  when 
log  (A  log  y)  is  plotted  to  log  x.  The  slope  gives  the  value  of 
c  and  the  intercept  gives  logbir^-i).  From  Fig.  i6  values 
of  b  and  c  are  readily  obtained. 


c  =  2%, 


&=— .02282. 

In  the  next  to  the  last  line  the  value  of  a  is  computed  for 
each  value  of  x  from  the  equation 

log  a  =  log  3;+.o228iC^*. 


52  EMPIRICAL  FORMULAS 

The  average  of  these  values  of  a  gives 

0  =  2.314. 
The  formula  obtained  is 

y  =  (2.3i4)io-«"«'". 

The  values  of  y  computed  from  this  equation  are  written 
in  the  last  line  of  the  table.    The  agreement  is  not  a  bad  one. 


CHAPTER  IV 
XVI.    {x+a)iy-\-b)=c. 
Points  represented  by  Ix—xt, I  He  on  a  straight  line. 

\        y-y^/ 

If  two  variables,  x  and  y,  are  so  related  that  the  points  repre- 
sented by  [x—Xic, ■  ]  lie  on  a  straight  line,  the  relation  between 

\         y-yJ 

the  variables  is  expressed  by  the  equation 
XVI  {x^-a){y+b)=c. 

'LQix—Xu  =  X, 

y-yfc=Y, 

where  Xk  and  yk  are  any  two  corresponding  values  of  x  and  y. 
From  the  above  equations 

x  =  X-\-Xk, 

y  =  Y-{-yt. 

Substituting  these  values  of  x  and  y  in  equation  XVI  we 
have 

{X-\-x,+a){Y-\-y,-\-b)=c, 
or 

XY+{y,-{-b)X+(xjc+a)Y+(x,+a)(yt+b)=c. 

Since  (xtj  yt)  is  a  point  on  the  curve 

(xic+a)(yt+b)=Cy 
and 

XY-{-(y,-j-b)X+(x,+a)Y=o. 

53 


54 


EMPIRICAL  FORMULAS 


Dividing  the  last  equation  by  Y 


X 


X-^{y,-]-hY-^-\-x,+a=o, 


or 


Y 


I     y    X  -\-a 


yk-\-b        y,-\-b' 


This  represents  a  straight  line  when  X  is  plotted  to  — . 

The  theorem  is  proved  directly  as  follows:    If  the  points 

(X  —  Xk\ 
x  —  x, I  lie  on  a  straight  line  its  equation  will  be 
y-yJ 


X-Xk 

y-y^ 


=  p{x-XK)+q. 


Clearing  of  fractions 

x-x:;  =  p{x-x.){y-yK)-{-q{y-yk)' 
This  is  plainly  of  the  form 

{x+a){y+h)=c. 

The  following  tables  of  values  is  taken  from  Ex.  i8,  page  138 
of  Saxelby's  Practical  Mathematics.  It  represents  the  results 
of  experiments  to  find  the  relation  between  the  potential  differ- 
ence V  and  the  current  A  in  the  electric  arc.  The  length  of 
the  arc  was  3  mm. 


A   (am- 

peres) 

1.96 

2.46 

2.97 

3.4s 

3.96 

4.97 

5.97 

6.97 

7.97 

V  (volts 

67.00 

62.7s 

59-75 

58.50 

56.00 

53.50 

52.00 

51.40 

50.60 

X 

0 

0.50 

1. 01 

1.49 

2.00 

3.01 

4.01 

5.01 

6.01 

Y 
X 
Y 

0 

-4.2s 

-7.25 

-8.50 

—  11.00 

-13.50 

—  15.00 

-15.60 

—  16.40 

-   .1176 

-    .1393 

-    .1752 

-      .1817 

—      .2228 

—      .2670 

-      .3210 

-      -366s 

Com- 

puted F 

66.99 

62.74 

59.80 

57.80 

56.19 

53-94 

52.44 

51.36 

50.55 

Let  A   be  taken  as  abscissa  and  V  as  ordinate  and  transfer 
the  origin  to  the  point  (1.96,  67.00)  by  the  substitution 

X=A-i.g6, 

7=7-67.00. 


DETERMINATION   OF   CONSTANTS 


55 


The  values  of  X  and  Y  are  given  in  the  third  and  fourth 

X 

lines  of  the  table.     The  values  of  —  are  plotted  to  X  in  Fig.  17 


3  4 

Yalues  of  X 


Fig.  17. 

and  are  seen  to  lie  nearly  on  a  straight  line.     It  is  therefore 
concluded  that  the  formula  is 

{V-Vh){A^a)=c. 

By  the  equations  of  substitution  this  becomes 

(X+i.96+a)(F+67.oo+6)  =c, 
or 

XY+{6T.oo+h)X-^{i.g6-\-b)Y=o. 

Dividing  by  7(67.00+^) 

1.96+d^ 


X 
Y 


__  X  _ 

67.00+6  67.00+6 


The  slope  of  this  line  is   — 

— — -.     From  Fig.  17 

67.00+6 


Solving  these  equations 


From  formula 


67.00+6 

1.96+a 
67.00+6 


67.00+6 

=  •045; 
=  .095. 


and  the   intercept    is 


^  =  0.151, 
6= -44.78, 

c  =  46.89. 


56  EMPIRICAL  FORMULAS 

These  values  give 

(A  +o.isi)(7-44.78)  =46.89. 

In  the  last  line  of  the  table  are  written  the  values  of  V  com- 
puted from  the  above  formula 

_b 
XYla.     y-=aio'-^'. 

Points  represented  by  ( log  —,  log  —  )  lie  on  a  straight  line. 

\x-xt        yt        yt/ 

If  two  variables,  x  and  y,  are  so  related  that  the  points  repre- 
sented by  I log  — ,  log  —  I  lie  on  a  straight  line,  the  relation 

\x-Xk      yk       ykj 

between  the  variables  is  expressed  by  the  equation 

b 
XYla  y=aio'+'. 

By  the  condition  stated 

log^  =  w  — ^  log^+^ 

yk        x-xt       yt 

where  Xt  and  yk  represent  any  two  corresponding  values  of 
X  and  y.  m  is  the  slope  of  the  line  and  b  its  intercept.  Clear- 
ing the  equation  of  fractions 

(log  y-log  yk){x-Xk)=m(}ogy-\og  yk)+b{x-Xk), 
or 

log  y{x-Xk-m)  =  (&+log  yk)x-\og  yk{xk-\-m)-bxt. 


iQg    _  (^+log  yQ:^ -log  yk{xk+m)  -bxi 


x—Xt—m 

^Ax+B 

x+C 

^^  x+C 

=logaH — — . 

x+c 


DETERMINATION   OF   CONSTANTS  57 

Therefore 

b 
y=joiogaj_QX+c 


For  the  purpose  of  determining  the  constants  the  equation 
is  written  in  the  form 

\ogy  =  \oga-\ — — , 

x-\-c 

(log  y -log  a){x+c)=b, 
Let 

log3;  =  logF+log3/fc, 
and 

x  =  X-^Xk. 
Then  follows 

(log  F+log  y,-\og  a){X+x,-hc)=b, 

X  log  Y  +log  Y(xic  +c)  +X(log  yic  -log  a)  +  (log  yt  -log  a)  (xt -\-c)  =  b. 

But 

(log  yic-log  a)(xk-\-c)  =b, 

since  the  point  {xk,  yk)  lies  on  the  curve. 

X  log  F+log  Y{xk+c) -\-X{log  yk-log  a)  =o. 
Dividing  this  equation  by  X 

log  Y=-(xk+c)  -^^+loga-log>'A;. 

Replacing  log  F  and  X  by  their  values 

log  ^  =  -  (xk+c)-^  log  — +log  a-log  yt. 
yk  x—Xk       yk 

From  this  it  is  seen  that  if  log  —  be  plotted  to  — ^ —  log  — 

yk  x-xt      yt 

a  straight  line  is  obtained  whose  slope  is  —{xk-\-c)  and  whose 
intercept  is  log  a— log  yk.  If  the  slope  of  the  line  is  represented 
by  M  and  the  intercept  by  B 

c=—M—Xij 

log  a  =  B -{-log  yk. 


58  EMPIRICAL  FORMULAS 

By  writing  XVIa  in  the  logarithmic  fonn 

\ogy  =  b—--\-loga 
a  line  is  obtained  whose  slope  is  b. 

XVII.    y  =  ae''-{-be'^. 

(y  t + 1  y* + 2\ 
} 1 
yt     yt  I 

lie  on  a  straight  line  whose  slope,  M,  is  positive  and  intercept,  5,  is  negative, 
and  lf2+45  positive. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of  x 
are   taken   in   an   arithmetical   series   the   points   represented   by 

(yk+\^yk+2\  1'^  ^^  ^  straight  line  whose  slope,  M,  is  positive 
yt     yt  I 

and  whose  intercept,  B,  is  negative  and  also  M^-\-4B  is  positive 
the  relation  between  the  variables  is  expressed  by  the  equation 

XVII  y  =  ae''-^be^. 

Let  {xk,yk),  (x-\-Ax,  yk+i),  fe+2Aa;,  yk+2)  be  three  sets  of 
corresponding  values  of  x  and  y  where  the  values  of  x  are  taken 
in  an  arithmetical  series.  We  can  then  write  the  three  equations, 
provided  these  values  satisfy  XVII. 

y,  =  a/^*+^>/^^       ........     (i) 

y.+i-a^^^V^^+5e^V^^ (2) 

Multiplying  (i)  by  e^^^  and  subtracting  the  resulting 
equation  from  (2) 

y,+i-e'^^y,  =  be^He'^^'-e'^'').    ....     (4) 

Multiplying  (2)  by  e^^^  and  subtracting  from  (3) 

Multiplying  (4)  by  e"^^^  and  subtracting  from  (5)  there 
results 


or 


DETERMINATION   OF   CONSTANTS  59 

3'.+2-(^^^"+e^^")^.-M+e(^+^^^^^.  =  o,.     .     .     (6) 

Jk  yt 

The  values  of  c  and  d  are  fixed  for  any  tabulated  function 
which  can  be  represented  by  XVII,   and  therefore,   the  last 

equation  represents  a  straight  line  when  ^^^^  is  plotted  to  ^— ^. 

yk  jfk 

The  slope  of  the  Hne  is 
and  the  intercept  is 

It  is  seen  that  M  is  positive,  B  negative,  and  M^+4B  posi- 
tive, for 

J|^2  =  ^2cAx_^^^(c  +d)Axj^^Ax^ 

and 

In  the  first  two  lines  of  the  table  are  given  corresponding 
values  of  x  and  y.  It  is  desired  to  find  a  formula  which  will 
ex'press  ^he  relations  between  them. 


y 

yk+i 

yk 
yk+2 

yk 

g-Al2x 
3.g-.165x 
Computed  y 


+  .3762 


+  .662 

+  •319 
+  .371 


1.5 


4- .0906 


+  .539 
+  .071 
+  .087 


-.1826 

+  .241 

-•48s 

+  ■439 
-131 
-.185 


2.5 


-  -4463 
-2.01S 

-4.926 

+   -359 

-  -295 

-  -447 


3.0 


-  -7039 

+2.444 

+3.855 

+   .290 

-  -429 

-  .704 


3-5 


-  .9582 

+1-577 

+2.147 

+   .236 

-  .538 

-  -957 


4.0 


— 1.2119 
+I.361 

+1.722 

+   .192 

-  .626 

—  1. 210 


4.5 


-1.4677 
+  1.265 

+I.S32 

+  •157 
-    .698 
-1.464 


5.0 


—  1.7280 
+I.211 

+  1.426 

+   .127 

-  .757 
-1.723 


Plotting  the  points  represented  by  (^^,  ^^ ),  Fig.    i8, 

\  yk      yt  / 

a  straight  line  is  obtained  whose  equation  is 


60 


EMPIRICAL  FORMULAS 


M=1.97, 

B=-  .96. 

Since  M  is  positive,  B  negative,  and  M^+4B  positive,  it 
follows  tiiat  tiie  relation  between  the  variables  is  expressed 
approximately  by  XVII.     It  has  been  shown  that  the  slope 


;? 

/ 

/ 

/ 

/ 

/ 

/ 

-2 
-3 
-4 

/ 

/ 

/ 

J 

Y 

-5 

/ 

>-A 


-1  0  1 

Values  of-^^ 

Fig.  18. 


.3         .4         .5         .6 
Values  of  e""*"* 
Fig.  19. 


of  the  line  is  equal  to  e'^'+e'^^,  and  the  intercept  is   equal  to 
_^(c+<n±c^     Since  A:)t:  is  .5 


e'''e''^  =  i.g'j, 


A<c+d)  _ 


.96. 


From  these  equations  are  obtained  the  values  of  c  and  d, 
c=-.247, 
^  =  .165. 


DETERMINATION   OF   CONSTANTS  61 

The  formula  is  now 

Dividing  both  sides  of  this  equation  by  e-^^^*  gives  the 
equation 

which  represents  a  straight  line  when  ye'-"^^^^  is  plotted  to 
^-.4121  jjjg  values  of  these  quantities  taken  from  the  table 
are  plotted  in  Fig.  19  and  are  seen  to  lie  very  nearly  on  a  straight 

line. 

This  line  has  the  slope  2.00  and  intercept  —  i.oi.  Sub- 
stituting these  values  of  a  and  b    in  the  formula  it  becomes 

It  is  seen  that  the  errors  in  the  values  of  y  computed  from 
this  formula  are  in  the  third  decimal  place.  The  values  are  as 
good  as  could  be  expected  from  a  formula  in  which  the  con- 
stants are  determined  graphically.  For  a  better  determination 
of  the  constants  the  method  of  Chapter  VI  must  be  employed. 

XVIII.     y  =  e^'ic  cos  hx-^d  sin  hx). 
Values  of  x  form  an  arithmetical  series,  and  the  points  ( — — ,  t^—1  \ 

\  yk     yk  / 

lie  on  a  straight  line.    Also  M^-\-/^B  is  negative. 

//  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  an  arithmetical  series  the  points  represented  by 

/Z£±l^  Z^iLj  11^  on  a  straight  line  whose  slope  M  and   intercept 

B  have  such  values  that  M^+4B  is  negative,  the  relation  between 
the  variables  is  expressed  by  the  equation 

XVIII  y  =  e^'ic  cos  bx-\-d  sin  bx). 

Let  X  and  y^  be  any  two  corresponding  values  of  the  variables. 
We  have  the  three  equations 

yk  =  6°" (c  cos  bx-\-d  sin  bx), (i) 

yk+i  =e°V^^[c  cos  (bx-\-bAx)-}-d  sin  (bx+bAx)] 


62  EMPIRICAL   FORMULAS 

=  c"e^'[c(cos  bx  COS  ftAa:  — sin  bx  sin  bAx) 

-\-d{sm  bx  cos  6Ax+cos  bx  sin  bAx)] 

=^ef'ef^[{c  cos  bAx+d  sin  6Aic)cos  6a; 

+  (</  cos  bAx  —  c  sin  6A:i:)sin  bx].     .     (2) 

The  value  yk+2  can  be  written  directly  from  the   value   of 
y..+ 1  by  replacing  Ax  by  2 Arr. 

3'.;+2  =  e°*e^*^'[(c  cos  26A:r+(/  sin  26Aic)cos  ftic 

+  (</cos  25Ajt:— c  sin  26A:j[;)sin6a;] (3) 

Subtracting    (i)    multiplied     by    e'^''{c  cos  bAx-\-d  sin  6Ait;) 
from  (2)  multiplied  by  c  we  have 

cyt+i  —  e"^^{c  cos  Ma;H-c?  sin  bAx)yk 

—  ce°^e'^^(d  cos  Mx— c  sin  6 Ax)  sin  bx 

— ^e°V^''(c  cos  bAx+d  sin  ft  Ax)  sin  Jx 

=  -(c2  +  (f2)gaa:gaAx5ijj^^^sin6x (4) 

Similarly 

cyk->r2  —  e^'^^{c  cos  2bAx-\-d  sin  26Ax)3;fc 

=  -  (<;2  +  C^2)gaXg2aAz  gj^  sZ^Ax  slu  6x.      .       .       .       .        (5) 

Multiplying  equation  (4)  by  e"^''  sin  2  6 Ax  and    subtracting 
it  from  (5)  multiplied  by  sin  bAx 

c  sin  bAxyk+2  —  e'^''^{c  cos  26AX  sin  bAx-\-d  sin  26AX  sin  bAx)yk 

—ce!^ sin  2bAxyi+i+e^''^''{c  cos  6Ax  sin  26AX 

+d  sin  bAx  sin  2Z>Ax);yi =0. 
Simplifying 

c  sin  bAxyk+2—cef^''  sin  26Ax3;a+i  H-^^^'*'^^  sin  ftAx^;*  =0. 

Dividing  by  c  sin  bAxy^, 

yf  yt 


DETERMINATION   OF   CONSTANTS 


63 


The  values  of  a  and  b  will  be  fixed  for  any  tabulated  function 
which  can  be  represented  by  XVIII,  and  therefore,  the  last 

equation  represents  a  straight  line  when  ^^  is  plotted    to 
^— ^.     The  slope  of  the  line  is 

yt 


and  the  intercept 


M  =  2e''^^  cos  JAiC, 


B  = 


It  is  evident  that  M^+4.B  is  negative. 

It  is  possible  that  in  a  special  case  M^+^B  might  be  zero, 
but  then  b  would  be  zero  and  hence 


y  =  ce''', 


which  is  formula  V. 

Corresponding  values  of  x  and  y  are  given  in  the  first  two 
columns  of  the  table  below.  It  is  required  to  find  a  formula 
which  will  represent  approximately  the  relation  between  them. 


y 

yjc+i 
yk 

yk+z 
yk 

^.OSx 

coshx 

tan  hx 

coshx 

y 

Com- 

X 

.•««^os  bx 

puted  y 

0 

+    .300 

I. 0000 

I 

.0000 

.0000 

I. 0000 

+    .300 

+    .308 

I 

+   .oil 



1.0833 

+ 

.8646 

+        .5812 

+  .9366 

+    .012 

+    .018 

2 

-    .332 

+  .04 

-  I. II 

1.1735 

+ 

.4950 

+     I. 7556 

+  .5809 

-    .571 

-    .327 

3 

-    .636 

-30.2 

-57. 8 

I. 2712 

— 

.0087 

-114-59 

—     .0111 

+57.3 

-    .634 

4 

—    .803 

+1.92 

+   2.42 

I. 3771 

- 

.5100 

-      1.6864 

-  .7023 

+   I. 143 

—    .804 

S 

-    .761 

+1.26 

+    1.20 

I. 4918 

— 

.8732 

.5581 

—  1.3026 

+      .584 

-    .761 

6 

-    .48s 

+    .95 

+      .60 

1.6161 

— 

.9998 

+        .0175 

-I.6IS9 

+      .300 

-    .485 

7 

-    .017 

+    .64 

+      .02 

1.7507 

- 

.8557 

+        .6048 

-1. 4981 

+      .011 

—    .012 

8 

+    .537 

+    .04 

-    I. II 

1.8965 

— 

.4797 

+      I. 8291 

-    .9098 

-      .590 

+    .545 

9 

+  1.027 

-31.6 

-60.4 

2.0544 

+ 

.0262 

—   38.1880 

+    .0538 

+  19.08 

+  I.03S 

10 

+  1.298 

+  I.91 

+   2.42 

2.2255 

+ 

.5250 

—      I. 6212 

+  1.1684 

+  1    .III 

+  1.299 

In  Fig.  20  the  points  represented  by  ( ^^,  ^^  )  are  plotted. 

\yt      yt  I 

They  lie  very  nearly  on  the  straight  line  whose  equation  is 

yk+2  o..-  yk+\ 

^  =  1.875^^- 1. 175. 

yji  jk 


64 


EMPIRICAL   FORMULAS 


Since   (1.878)2—4(1.18)   is  negative  the  relation  between    the 

variables  is  expressed 
approximately  by  the 
equation 

y  =  €"^{0  cos  hx-\-d  sin  hoo). 

It  was  shown  that 
the  slope  of  the  line  is 
equal  to  2  (cos  hb.oc)e'"^'' 
and  the  intercept  equal 
to  — ^^'"^*.  Since  ^x  is 
equal  to  unity  we  have 

2e°  cos  J  =  1.875, 

^'''  =  1.175, 


/ 

/ 

Q  A 

/ 

/ 

/ 

1^ 

/ 

/ 

1  n 

/ 

/ 

;^< 

/ 

% 

J 

/ 

1  •' 

/ 

«  .1 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

-1.2 

/ 

A      .6      .8      1.0    1.2    1.4     1.6    1.8    2.0 
Values  of  ^7^- 


FlG.  20. 


log  COS  6  =  9.9370—10, 

6  =  30°  10'  ap- 
proximately, 

a  =  .08. 

The  formula  is  now 


y  =  e-^^^{c  cos  2P\x-\-d  sin  30^0;), 

where  30g^  is  expressed  in  degrees. 

Dividing  the  equation  by  c-^"^  cos  2Pk^ 


-^ — ^ — — 3  =  c+(/  tan  3oiic°, 
e-"^*  cos  ^o^x 


which  is  a  straight  line  when  —^^ — -—^  is  plotted  to  tan 

^  e-""^^  cos  zokx 

30^0;°.     In  Fig.  21  these  points  are  plotted  and  are    seen  to 

lie  nearly  on  a  straight  line  whose  slope  is  —  .496  and  intercept 


DETERMINATION   OF   CONSTANTS 


65 


.308.  Two  of  the  points  are  omitted  in  the  figure  on  account 
of  the  magnitude  of  the  coordinates.  Substituting  the  values 
of  constants  just  found  in  the  formula  the  equation  expressing 
the  relation  between  x  and  y  is 

^:«:g08^(.3o8  cos  3oir*:°  — .496  sin  3oix°). 


The  last  column 
in  the  table  gives 
the  values  of  y 
computed  from  the 
equation.  The 
agreement  with  the 
original  values  is 
fairly  good. 

In  case  c  is  zero 
XVIII  becomes  the 
equation  for  damped  vibrations,   y  =  J^°^  sin  bx. 


1.5 

r 

^2  0.5 

"~" 

~~" 

^ 

-^ 

'v, 

», 

""v 

"^ 

^ 

n. 

^ 

03 

, 

>!  -0.5 

^ 

\ 

N 

-1.5       -1.0      -0.5  0  0.5 

Values  of  tan  30Hx° 

Fig.  21. 


2..0 


XIX.     y  =  ax'^+bx^. 
Values  of  x  form  a  geometrical  series,  and  the  points 


yk+2\ 

yic  / 

lie  on  a  straight  line,  whose  slope,  M,  is  positive,  and  whose  intercept, 
B,  is  negative,  and  M^+4B  positive. 


.  ^  fyt+i: 

ints  I ,- 

\  yt 


If  two  variables,  x  and  y,  are  so  related  that  when  values  of 
X  are  taken  in  a  geometrical  series  the  points    represented    by 

j*-i-i^^fc+2\  ^^.^  ^^  ^  straight  line  whose    slope,  M,  is   positive, 
yk     yic  I 
and  intercept,  B,  negative,  and  also  M^-\-^B  positive,  the  relation 
between  the  variables  is  expressed  by  the  equation 

XIX  y  =  ax'+bx''. 

Let  X  and  y^  be  any  two  corresponding  values  of  the  variables. 
The  following  equations  are  evident: 

yt  =  ax''-\-bx^, (i) 


yt+i=ax''r^-\-bx^r^ 


(2) 


66  EMPIRICAL   FORMULAS 

yk-r2  =  axfr'''-\-bx'r^'', (3) 

yi.^i-r'y,    ^hx^r'-r^), (4) 

yk+2-r'yt+i  =bxfr^{r^-r') (s) 

Multiplying  equation  (4)  by  y'^  and  subtracting  it  from 
equation  (5)  there  results 

yt+  2  -  r'yt+ 1  -  r^yk+ 1  +r'-^'^yt  =  o, 
or 

yt  yt 

It  is  seen  that  the  slope  of  this  line  is  positive  and  the  inter- 
cept negative,  and  M^+4B  positive. 

In  the  table*  below,  the  values  of  x  and  y  from  a;  =  .05  to 
x  =  .ss  are  taken  from  Peddle's  Construction  of  Graphical 
Charts. 


yk+i 

yk-¥i 

.55 

85 

y 

Com- 

X 

y 

X 

y 

yk 

yk 

^.00 

r°'' 

x'' 

puted  y 

•OS 

.283 

■OS 

.283 



192 

.078 

1.470 

.283 

10 

.402 

.10 

.402 

282 

.141 

1.426 

.402 

15 

.488 



352 

.199 

1-385 

.488 

20 

.556 

.20 

.556 

1.420 

1.965 

413 

.255 

1-347 

.556 

25 

.613 

466 

.308 

I-315 

.612 

30 

.658 



516 

•359 

1.276 

.658 

35 

•  695 

561 

.410 

1.238 

.697 

40 

•  730 

.40 

•  730 

1-383 

1. 816 

609 

-459 

1.208 

.730 

45 

•757 

645 

•507 

1. 174 

.757 

50 

.780 

683 

-555 

1. 142 

.780 

55 

.800 

_ 

720 

.602 

1. 114 

.799 

60 

.814 

755 

.648 

1.078 

.814 

65 

.826 

789 

-693 

1.047 

.826 

70 

.835 

822 

-738 

1. 016 

•83s 

75 

.840 

.854 

.783 

0.984 

.840 

.80 

.845 

.80 

.845 

1-313 

1.520 

.885 

.829 

0.955 

.846 

In  column  3  the  values  of  x  are  selected  in  geometrical  ratio 
and  the  corresponding  values  of  y  are  given  in  column  4.    The 

points  /Z^±1^2!^j  are  plotted  in  Fig.  22,  and    although    the 

\  yt    yt  / 

*  See  Rateau's  "  Flow  of  Steam  Through  Nozzles." 


DETERMINATION   OF   CONSTANTS 


67 


three  points  do  not  lie  exactly  on  a  straight  line  the  approxima- 
tion is  good.  The  slope  of  the  line  is  4.10  and  the  intercept 
—3.86  which  give  the  equations 

2^+2^=4.10, 

2'=+'^  =  3.86. 


1.60 

Values  of  a;-*^ 
.10    .20     .30     .40     .50     .60    .70     .80     .90 

1.50 
iol-40 
^1.30 
|l.20 
^  1.10 

\ 

N 

\ 

N 

X 

N| 

H 

1  00 

X 

Si 

.90 

Sj 

N 

N 

V 

N 

\ 

/ 

N 

\/ 

/ 

/ 

\ 

N. 

/ 

/ 

/ 

/' 

/ 

/ 

/ 

/ 

/ 

/ 

2.0O 


1.80^1" 


1.70^ 


1.60 


1.30  1.32  1.34  1.36  1.38  1.40  1.4 

Values  ol  ^p- 

Fig.  22  AND  Fig.  23. 

Solving  these  equations  the  values  of  c  and  d  are  found  to  be 
e  =  i.40, 

<^  =  -5S- 
The  formula  now  is 


68  EMPIRICAL   FORMULAS 

Dividing  both  members  of  this  equation  by  r" 

which  represents  a  straight  line  when  -^  is  plotted  to    rc^*. 

The  slope  of  the  line  is  equal  to  a  and  the  intercept  equal  to  b. 
From  Fig.  23 

a=-  .685, 

6  =  1.522. 

The  formula  after  the  constants  have  been  replaced  by  their 
numerical  values  is 

The  last  column  of  the  table  shows  that  the  fit  is  quite 
good. 

If  the  errors  of  observation  are  so  small  that  the  values 
of  the  dependent  variable  can  be  relied  on  to  the  last  figure 
derivatives  may  be  made  use  of  to  advantage  in  evaluating  the 
constants  in  empirical  formulas.  But  when  the  values  can  not 
be  so  relied  on,  or  when  the  data  must  first  be  leveled  graphically 
or  otherwise,  the  employment  of  derivatives  may  lead  to  very 
erroneous  results.  This  will  be  illustrated  by  two  examples 
worked  out  in  detail. 

The  first  step  in  the  process  is  to  write  the  differential  equa- 
tion of  the  formula  used  and  then  from  this  equation  find  the 
values  of  the  constants. 

Consider  the  formula 

y  =  e"'(c  cos  bx-\-d  sin  bx). 

Looking  upon  a  and  b  as  known  constants  and  c  and  d 
as  constants  of  integration,  the  corresponding  differential 
equation  is 

Dividing  this  equation  by  y 

y      y 


DETERMINATION   OF   CONSTANTS 


69 


which,  if  the  data  can  be  represented  by  XVIII,  represents  a 
straight  Hne  whose  slope  is  2a  and  whose  intercept  is  —  (a^+ft^). 
Corresponding  values  of  x  and  y  are  given  in  the  table. 


. 

y' 

y" 

yl 

yl 

X 

gMx 

cos  .08a: 

tan  .08a; 

y 

X 

g.OCx 

y 

y 

De- 

Min- 

cos .08a: 

grees 

utes 

0 

+ 

.3000 

0 

0 

I. 0000 

I. 0000 

.0000 

+ 

.3000 

J 

+ 

.2750 

4 
9 

35.02 
10.04 

I  0618 

.9968 
.9872 

0802 

+ 
+ 

.2598 

.2193 

2 

+ 

.2441 

-.02,\2 

-.0068 

_ 

.1401 

-.0279 

I. 1275 

.1614 

3 

+ 

.2065 

—  .043() 

-.0066 

— 

.197b 

-.0319 

13 

45.06 

I. 1972 

.9713 

.2447 

+ 

.1776 

4 

+ 

.1622 

-.0481 

-.0078 

— 

.  29bS 

—  .0481 

18 

20.08 

I. 2712 

.9492 

.3314 

4- 

.1344 

S 

+ 

.1102 

-.0557 

-.0075 

— 

.5o,=;4 

-.0681 

22 

55. 10 

1.3499 

.9211 

.4228 

+ 

.0886 

6 

+ 

.0506 

-.0635 

-.0086 

—  I 

.2549 

—  .1700 

27 

30.12 

1-4333 

.8870 

.5206 

+ 

.0398 

7 

— 

.0175 

-.0721 

- . 0080 

+4 

.1200 

+  .4571 

32 

05.14 

1.5220 

.8472 

.6270 

.0136 

8 

— 

.0937 

-.0805 

-.0087 

+ 

.8591 

+  .0928 

36 

40.16 

1.6161 

.8021 

.7446 

— 

.0723 

9 

— 

.178b 

-.0894 

—  .0091 

+ 

.5011 

+  .0510 

41 

15.18 

I. 7160 

.7519 

.8771 

— 

.1384 

10 

— 

.2726 

-.0985 

—  .0091 

+ 

.3247 

+  .0334 

45 

50.20 

I. 8221 

.6967 

1.0296 

— 

.2147 

II 

— 

■^iv 

-.1078 

-.0085 

+ 

.2869 

+  .0226 

50 

25.22 

1.9348 

.6372 

I . 2097 

— 

.3047 

12 

— 

.4881 

-.1168 

-.0087 

+ 

.2393 

+  .0178 

55 

00.24 

2.0544 

.5735 

1.4284 

— 

.4143 

i,^ 

— 

.bo93 

-.1257 

—  .cx>9i 

+ 

.2063 

+  .0149 

59 

35.26 

2.1815 

.5062 

I . 7036 

— 

.5518 

14 

— 

.739b 

-.1348 

-.0089 

+ 

.1823 

+ . 01 20 

64 

10.28 

2.3164 

.4357 

2.0659 

— 

.7328 

IS 

— 

.8788 

-1435 

-.0084 

+ 

•  Ib33 

+.0096 

68 

45.30 

2.4596 

•3^24 

2.5722 

— 

.9859 

Th 

73 
77 

20.32 
55-34 

2.6117 
2.7732 

.2867 
.2098 

3.3414 
4.6735 

-3707 
.0306 

—  I 

.1814 

—2 

The  values  of  y'  and  y"  are  obtained  by  the  formulas 

y'n=—r      (>-2-8>'„-i+  ^yn+i-yn+2), 
i2n 

y"n=  -^^  {yn-2-i(>yn-i+2>oyn-i^yn+i+yn+2)y 

where  /f  =  Ax  =  i .     These  formulas  are  derived  in  Chapter  VI. 
Plotting  the  points  represented  by  (  —  ,—),  Fig.  24,  it  is 

v^'    y  / 

seen  that  they  lie  nearly  on  a  straight  line  whose  slope  is  .12 
and  intercept  —.01.     Therefore 

2a  =  .i2,' 

a2-f52  =  .OI, 

a  =  .06, 

6  =  .08. 
We  have  then 

y  =  e-^^^{c  cos  .o8x+J  sin  .o8x). 


70  EMPIRICAL  FORMULAS 

Dividing  this  equation  by  e^**  cos  .oSx 

oto   ^ — ^  =  c-\-d  tan  .08a;. 
e""*  cos  .080; 


-1.2  -1.0    -. 

8     -. 

6      -.4     -. 

2      - 

w 

1  . 

4 

6       . 

8      1.0 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

A 

/ 

/ 

/ 

r 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

A 

Fig.  24. 


This  represents  a  straight  line  when 


r,.    ^ — —  is  plotted 
e-^^^  cos  .oSo;       ^ 

to  tan  .08a:.    The  slope  is  d  and  intercept  c.    From  Fig.  25 

c=     .3. 
The  law  connecting  the  variables  is  represented  by 
y  =  er^^^{^,2,  cos  .oSic  — .5  sin  .o8x). 


DETERMINATION   OF   CONSTANTS 


71 


The  values  of  y  computed  from  this  formula  agree  with 
those  given  in  the  second  column  of  the  table. 
Consider  formula  XIX 

The  corresponding  differential  equation  is 

y  y 


2  3 

Values  of  tan  Mx 


M 

■eO.C:r 

cos 

\ 

\ 

\, 

\ 

\ 

\ 

\ 

\ 

\, 

\ 

\ 

\ 

\ 

1 

\ 

\ 

1 

\ 

\, 

N 

\ 

\ 

\, 

\ 

\ 

\ 

V 

A 

.2 
0> 
-.2    o 


-.6 


-8 

-1.2 
-1.4 
.6 
-1.8 
-2.0 


Fig.  25. 

where  c  and  d  are  known  constants  and  a  and  h  constants  of 
integration.     The  differential  equation  represents  a  straight  line 

The  slope  is  c-\-d—i  and  the  inter- 


0(P"V  XV 

when  —^—  is  plotted  to  — 

X  V 


cept  is  —cd. 

The  values  of  x  and  y  in  the  table  below  are  the  same  as 
those  given  in  the  discussion  of  XIX. 


72 


EMPIRICAL  FORMULAS 


/ 

x/ 

x^" 

X 

y 

y 

'" 

y 

y 

•OS 

.283 

lO 

.402 

IS 

.488 

1.503 

-6.933 

.462 

-.320 

20 

•  556 

1.240 

-4.133 

•  446 

-.297 

25 

•  613 

1. 015 

-4.967 

.414 

-.506 

30 

.6s8 

0.803 

-3.267 

.366 

-.447 

35 

.695 

0. 720 

—0.400 

•  363 

-.071 

40 

.730 

0.623 

-3.533 

.341 

-.774 

45 

•757 

0.492 

—  1.500 

.292 

-.401 

SO 

.  .780 

0.433 

—  1 . 067 

.277 

-.342 

55 

.800 

0.338 

-2.633 

.232 

-.996 

60 

.814 

0.2S5 

-0.633 

.188 

-.280 

65 

.826 

0.213 

-1.200 

.167 

-.614 

70 

.835 

0^135 

-1.767 

•  113 

-1.037 

75 

.840 

80 

.845 

The  values  of  y'  and  y"  were  computed  by  means  of  the 
formulas  used  in  the  preceding  table.     In  Fig.  26  the  points 

/  _Z^  _J_  J  are  plotted  and,  as  is  seen,  the  points  do  not  deter- 
mine a  line.  It  is  clear  that  the  constants  can  not  be  determined 
by  this  method. 

XlXa.    y^aofcf. 

Points  represented  by  ( Xn,  log  — —  )  lie  on  a  straight  line. 
\  yn  / 

If  two  variables,  x  and  y,  are  so  related  that  when  values  oj  x 
are  taken  in  a  geometrical  series  the  corresponding  values  of  y  are 

such  that  the  points  represented  by  I  Xn,  log  ■^^^  ]  lie  on  a  straight 

\  yn  J 

line,  the  relation  between  the  variables  is  expressed  by  the  equation 
XrXa  y  =  ay^(f. 

Using  logarithms: 

log  yn  =  l0g  a-\-b  log  Xn-\-Xn  log  C, 

log  y„+i  =log  a-\-b  log  Xn-\-rXn  log  c+6  log  r. 


DETERMINATION   OF   CONSTANTS 


Subtracting  the  first  equation  from  the  second 


73 


log 


yn+i 


{r-i)xn\ogc+b  logr. 


By  plotting  log  ^^  to  o^  a  line  is  obtained  whose  slope  is 

*  .     -^"  . 
(/•  —  i)log  c,  and  since  r  is  known  c  can  be  determined. 


° 

. 

° 

o 

g-6 

^ifi 

o 

° 

Values  of  -jp 
Fig.  26. 

From  the  first  equation 

log  yn  -  Xn  log  c  =  I)  log  :^„ +log  a. 

If  then  log  >>«  — it:„  log  c  be  plotted  to  log  Xn  a  line  is  obtained 
whose  slope  is  b  and  whose  intercept  is  log  a. 


CHAPTER  V 

XX.    y=Oo+fli  cosx+oacos  2x+a3  C0S3X+  .  .  .  -\-aTCosrx 

-\-b1sinx-\-b2sm  2X+b3sin$x-\-  .  .  .  +6r_iSm  (r— i)x. 
Values  of  y  periodic. 

The  right-hand  member  of  XX  is  called  a  Fourier  Series 
when  the  number  of  terms  is  infinite.  In  the  application  of  the 
formula  the  practical  problem  is  to  obtain  a  Fourier  Series, 
of  a  limited  number  of  terms,  which  will  represent  to  a  sufficiently- 
close  approximation  a  given  set  of  data.  The  values  of  y  are 
given  as  the  ordinates  on  a  curve  or  the  ordinates  of  isolated 
points. 

In  what  follows  it  is  assumed  that  the  values  of  y  are  periodic 
and  that  the  period  is  known. 

We  will  determine  the  constants  in  the  equation 

y  =  ao+ai  cos  x-\-a2  cos  2x+as  cos  3a;,-f&i  sin  x-\-b2  sin  2x, 

so  that  the  curve  represented  by  it  passes  through  the  points 
given  by  the  values  in  the  table. 


X 

0° 

60° 

120° 

180° 

240° 

300° 

360° 

y 

I.O 

1-7 

2.0 

1.8 

1-5 

0.9 

1.0 

Substituting  these  values  in  the  equation  we  have  the  fol- 
lowing six  linear  relations  from  which  the  values  of  the  six 
constants  can  be  determined: 

i.o  =  ao+  fli+  (i2-\-a3y 

i.7  =  ao-\-^ai-la2-a3-\ — -h-\ ^62, 

2  2 

2.o  =  ao-iai-^a2^a3^ — ^^i ^62, 

2  2 

74 


DETERMINATION   OF   CONSTANTS  75 

i.S  =  ao—  ai+  a2—az, 

i.S  =  ao-^ai-^a2+a3 ^61 H — ^^>2, 

2  2 

2  2 

Multiplying  each  of  the  above  equations  by  the  coefficient 
of  ao  (in  this  case  unity)  in  that  equation  and  adding  the  result- 
ing equations  we  obtain  (i)  below.  Multiplying  each  equation 
by  the  coefficient  of  ai  in  that  equation  and  adding  we  obtain 
(2).  Proceeding  in  this  manner  with  each  of  the  constants 
a  new  set  of  six  equations  is  obtained. 

6^0  =  8.9. (i) 

3ai=-i.25.     .......  (2) 

302= -.25 (3) 

6^3  =  . 10 (4) 

36i  =  .65V3. (5) 

3^2  =  . 15  v"! (6) 

The  equation  sought  is 

>'  =  f§— A  cos  :x:-tV  cos  2x+-h  cos  2>^-]-¥Wi  sin  x+^V3  sin  2X. 

It  reproduces  exactly  each  one  of  the  six  given  values. 

The  solution  of  a  large  number  of  equations  becomes  tedious 
and  the  probability  of  error  is  great.  It  is,  therefore,  very 
desirable  to  have  a  short  and  convenient  method  for  com- 
puting the  numerical  values  of  the  coefficients.* 

*The  scheme  here  used  is  based  upon  the  12 -ordinate  scheme  of 
Runge.  For  a  fuller  discussion  see  "A  Course  in  Fourier's  Analysis  and 
Periodogram  Analysis  "  by  Carse  and  Shearer. 


76 


EMPIRICAL  FORMULAS 


Take  the  table  of  six  sets  of  values 


X 

o° 

60° 

120° 

180^ 

240° 

300° 

y 

yo 

yi 

y2 

3^3 

^4 

ys 

where  the  period  is  27r. 

For  the  determination  of  the  coefficients  the  following  six 
equations  are  obtained: 

2  2 

3'2  =  ao-Jai-§a2+fl^3H — ^^1 ^^2, 

2  2 

2            2 
3'5  =  ao+Jai-§a2-a3 ^^1 ^62. 

2  2 

Proceeding  in  the  same  way  as  was  done  with   numerical 
equations  the  following  relations  are  obtained: 

6ao  =  >'o+       yi-\-      3'2+3'3+       >'4+  yb, 

3«i=3'o+    \yi-    hyz-ys-    iy4.-\-hy5, 


3a2=yo-    iyi-    iy2+y3-    h^-iy5, 


6a3=yo 
3bi  = 

3^2 


yi+     >'2-3'3+     y^-  y5, 
V2     ,  V2  V2       V2 

22  2  2 

V2  Vt.  ,    \/2  V2 

2  2  2  2 


(a) 


For  convenience  in  computation  the  values  of  y  are  arranged 
according  to  the  following  scheme: 

yo         yi  y2 

ys n 3^ 

Sum  2^0  vi  V2 

Difference  wo  wi  W2 


DETERMINATION   OF  CONSTANTS 


77 


vo 


V2 


Wo 


Wi 

W2 


Sum 
Difference 


po 


Pi 


Sum 
Difference 


ro 


ri 

Si 


ib) 


6ao  =  po-\-pi,       1 
Sai=ro-\-isi, 

Sa2=po  —  hpij 
6a3  =  ro-si, 

2 


It  is  evident   that   the   equations   in  set  (6)  are   the   same   as 
those  in  set  (a). 

For  the  numerical  example  the  arrangement  would  be  as 
follows: 


i.o 

1.8 


1-7 

1-5 


2.0 
0.9 


Wo 


2.8 


2.8 

-.8 
3-2 

2.9 


3-2 

.2 


2.9 

I.I 
-.8 


.2 

I.I 


po 


2.8  6.1  n 

•3  Si 

6^0=4-8.90, 
3^1  =-1.25, 
3^2=-  .25, 
6^3  =  +  .10, 
Sbi  =  +  .65  V3, 

362  =  +  .i5V'3. 


-.8 


1-3 
-•9 


78  EMPIRICAL  FORMULAS 

It  is  seen  that  the  computation  is  made  comparatively  simple. 
The  values  of  the  v's  are  indicated  by  vo,  the  first  one.  The 
values  of  the  />'s,  etc.,  are  indicated  in  the  same  way. 

8-ORDiNATE  SCHEME.  The  formula  for  eight  ordinates  which 
lends  itself  to  easy  computation  is 

y  =  ao+cii  cos  d-\-a2  cos  2d-{-az  cos  sd+a^  cos  46 
+61  sin  d+b2  sin  2^+&3  sin  3^. 

For  determining  the  values  of  the  constants  eight  equations 
are  written  from  the  table : 

O     45°     90°     135°    180°     225°     270°   315° 


yo      yi         y2         ys         y^  ys  ye        yi 

yo  =  ao-\-      ai+fl2+      ^3+04, 

^2  ,   ^^2,      ,   ,      ,    V2, 

222 
I  ^2  ,  V2,      7    ,  V2  , 

H ^3  — ^4H Oi—02-\ 2O3, 

2  2.2 

—  03  +  ^4, 

V^2  ^^2  ^^2  V2 

y5  =  ao fli        H as  —  dA 61+^2 ^3, 

2  2  22 

y6=00  — <3^2  +^4  —  ^1  +  ^3, 

\/2  "^^2  V  2  V  2 

y7  =  floH <ii 0,3  — a^ hi— 1)2 63. 

22  22 

From  which  are  obtained  the  following  eight  equations: 

8flo=yo+     yi+y2+     y3+y4+     yb^-y&-{-     y?, 


yi 

=  ao-f 

V2 

a\ 

2 

y2 

=  ao 

-a2 

yz- 

=  ao- 

V~2 

ai 

2 

y^ 

=  aQ- 

ai- 

fa2 

4<3^i=yo+     yi 

V2             V2           ,  V2 

y3    y4         ys       +  ^  3^7 
222 

4^2 =yo 

-y2           +y4           -ye, 

4^3  =yo     ^  yi 

,  \/2             ,  V2               V2 

+  ^  y3    y4+  ^  ys           ^  yr 

DETERMINATION   OF   CONSTANTS  79 


Sa^=yo- 

yi+y2-      yz+y^ 

i-     y5+y6-     yi, 

4^1  = 

V2       ,         ,  \/2 

— yi+y2-\ ys 

2                2 

V2                      V-2 

^y5    ye     ^  yi, 

4h  = 

3^1      -     ys 

-\-     y^      -     yi, 

4^3  = 

V2          ,  V2 
— yi-y2-\ ys 

V2     ,        V2 
2                2 

For  the  purpose  of  computation  the  values  of  y  will  be 
arranged  as  follows: 

y^  y\          y2          yz 

3'4  y-o           yo,           yi 


Sum 

^0 

Z^l                    2^2 

Z'S 

Difference 

Wq 

Wl                  W2 

7£;3 

X'O 

^1 

ze;o 

Wl           W2 

V2 

2^3 

Wz 

Sum 

^ 

^1        Sum 

ro 

ri         r2 

Difference 

?o 

q\         Difference 
2 

4^2=^0, 

4<^3=?'o, ^1, 

2 

Sa4.=po—pi, 

^0      ■ 

Sl 

-'       V2      . 
4^1  = ''2  H n, 

^\-|-W 

-t  ^^j  i 

4^2 =?i, 

463= -/'2+      n. 

'^-^ 

+  ■ 

The  process  will  be  made  clear  by  an  example : 


0 

45° 

90° 

135° 

180° 

225° 

270° 

315° 

360° 

4 

—  2 

—  I 

2 

3 

3 

—  I 

2 

4 

80 


EMPIRICAL   FORMULAS 


For  computation  the  arrangement  is  as  follows: 
4—2—12 
3  3-12 


vo 

7 

I 

—  2 

4 

Wo 

I 

- 

-5 

0 

0 

7 

I 

I 

-5            0 

—  2 

4 
5 

ro 

0 

Po          5 

I 

-5            0 

^0          9 

-3 

Sao  = 
4(11  = 

4^2  = 
4^3  = 

8a4  = 

4^1  = 
4^2  = 

4&3  = 

10, 

i-fv;, 

9. 

i+fv;, 
0, 

-3, 

-5 

The  formula  becomes 

>;  =  i.25- 

■.634^ 

cos  d-\- 

2.25  cos  2 

^+1 

.134  cos  3^     ' 

— 

■.884: 

sin0- 

.75  sin  2( 

?-   . 

884  sin  3(9. 

10 

-Ordinate  Scheme 

yo 

yi 

y2 

3'3 

3'4 

yo 

ys 

3'7 

Jo 

J5 

Sum               I'o 

Vl 

V2 

7'3 

1)4. 

2'6 

Difference 

Wi 

W2 

2£;3 

W4. 

Vo 

Vl 

V2 

2e;i          «;2 

V5 

V4: 

vz 

Sum 

w;4         wz 

Sum                 po 

pi 

P2 

h           I2 

Difference        ^0 

91 

q2 

Difference       ;^i         W2 

DETERMINATION   OF   CONSTANTS 


81 


ioao  =  po-\-pi+p2, 
S(ii=qo-^Ciqi-\-C2q2, 
5^2  =pQ+C2pi—Cip2, 
5^3  =qo-  C2qi  —  Ciq2, 
5^4  =  po  —  Cipi  -\-C2p2j 

ioa5  =  qo  —  qi  +  q2, 
5h=Sili+S2l2, 
Sb2=S2mi+Sim2, 
Sb3=S2h—Sil2, 

5&4  =  'S'imi— 6*2^2. 


In  the  above  equations 

Ci  =  cos36°, 
C2  =  cos  72°, 


5i=sin36°, 
52  =  sin72°. 


In  the  schemes  that  follow,  as  in  the  lo-ordinate  one,  only 
the  results  will  be  given. 


12-Ordinate  Scheme 


yi 

) 

yi 

yn 

y2 
yiQ 

ys 

y9 

y4.       y5 

ys       yi 

y^ 

Sum 

Vq 

Vl 

V2 

Vs 

•d^        V5 

•VQ 

Difference 

Wi 

W2 

W3 

W^           W5 

vo 

n 

V2 

V3 

Wi 

W2      W3 

Vq 

V5 

n 

Sum 

W5 

W4. 

Sum 

po 

Pl 

P2 

pz 

ri 

r2     rz 

Difference 

qo 

qi 

q2 

Difference  si 

S2 

pc 

) 

pl 

r\        qo 

ll 

Ps 

rz         q2 

Sum 


Difference 


t2 


82  EMPIRICAL  FORMULAS 

I2flo  =  /o+/l, 

2 

6a2=Po—p3+^(pi—P2)f 

6a4^po-hp3-'i{pi+p2), 

2 

I2fl6=/o  — /l, 

6bi=iri-\ — ^r2+r3, 
2 

662=— ^(51+52), 

2 


663=/!, 

664=— ^(^1-^2), 
2 

6^>5  =  Jn ^^2  +  ^3. 


16-ORDINATE  Scheme 

yo      yi      y2      ys      y^     y^      y^      yi     ys 
y\b     yu     yi3     712    yn     yio     yo 


Sinn                 z;o 

n 

V2 

vs 

n 

V5 

I'd          V7 

V8 

Difference 

Wi 

W2 

Ws 

W4: 

W5 

We      W7 

vo 

Vl 

V2 

V3 

V4: 

•V8 

V7 

Vd 

V5 

Sum 

po 

pi 

P2 

Ps 

P4. 

Difference 

go 

qi 

Q2 

Qs 

DETERMINATION  OF  CONSTANTS  83 

Wl  W2.  Ws  W4: 

W7  Wq  W5 


Sum  ri         r2         rs  r^ 

Difference  ^i         S2         sz 

po      pi  p2  lo        h 

pA      p3  h 


Sum  lo       h       h  Sum  to 

Difference       mo     mi  Difference        xq 

i6ao  =  /o+/i, 

Sai=qo-\ g2+Cigi+C25'3, 

2 

V2 

2>a2  =  mo-\ Wl, 

2 

8^3  =  ^0 92  —  Ci^3  +C2^1, 

8^4=^0  J 

8^5  =  qo ?2 +Ci^3 — C2g'x, 

2 

8^6  =  Wo mi, 

2 

V2 

8^7 = go  H ^2 — Ci^i — C2g3 , 

2 

i6a8  =  ^— /i, 

V2 

86i  =  r4H r2+Cir3+C2ri, 

2 

\/2 

862=^2H (^1+^3), 

2 

863=  -^4  H ^2+Cin-^2;'3, 

2 

8^4  =  ^1— -^3, 


84 


EMPIRICAL  FORMULAS 


865  =  r4 r2+Ciri  -C2r3, 

2 

8^6=  -^2  H (^1+53), 

2 

V~2 

^h  =  -r^ r2-\-Cirz-\-C2ri. 


Ci  =  cos  22^°  =  sin  67!°, 
C2  =  sin  22|°  =  cos675°. 


20-Ordinate  Scheme 

yo    y\    y2    yz    y^    yb    y&    yi    ys    y^    y\z 
3'i9  yis  yn  y\(s  yib  yi4.  yiz  yi2  yn 


Sum 

vo 

Vl 

V2        Vz 

V4: 

1)5 

Vq 

V7        V8 

Vq      2;io 

Difference 

, 

Wi 

W2      W5 

\           W4: 

W5 

Wq 

W7      Wg      Wq 

Vo 

n 

V2 

Vz 

V4. 

■V5 

^10 

Vd 

V8 

V7 

Vo, 

Sum 

Po 

pi 

P2 

Pz 

P^ 

P5 

Difference 

qo 

Qi 

q2 

?3 

?4 

• 

Wi 

W2 

Wz 

' 

W4. 

W5 

, 

W9 

Ws 

Wi 

We, 

Sum 

ri 

r2 

rz 

ta 

Tb 

Difference 

Sl 

S2 

sz 

S4. 

po 

Pi 

P2 

qo 

qi 

q2 

P5 

P^ 

pz 

^4 

qz 

Sum 

h 

h 

12 

Sum 

ko 

ki 

k2 

Difference 

mo 

h 
h 
h 

mi 

W2 

Wo 

W2 

W] 

L 

ri        rz 

Sum 

to 

Sum 

no 

m 

Sum 

Ol           Oz 

DETERMINATION   OF   CONSTANTS  85 


Si 

S2 

^4 

S3 

^1 

g2 

hi 

h2 

Sum 
Difference 

2oao  =  to, 

ioai=qo-j-qi  sin  72°+g2  sin  54°+^3  sin  36°+g4  sin  i8°, 

ioa2  =  mo+mi  sin  54°+W2  sin  i8°, 

ioa3  =  qo  —  qs  sin  72°  — g4  sin  54°+gi  sin  36°  — 5^2  sin  18°, 

10^4  =  /o  — ^2  sin  54°+/i  sin  18°, 

10^5  =  ^0  —  ^2, 

ioaQ  =  mo  —  fn2  sin  54°  — mi  sin  18°, 

ioa7  =  qo+qs  sin  72°  — ^4  sin  54°  — ?i  sin  36°  — ^2  sin  18°, 

ioa8  =  /o  — ^1  sin  54°+/2  sin  18°, 

10^9  =  ^0—^1  sin  72°+?2  sin  54°  — ^3  sin  36°+g4  sin  18°, 

20^10=^0  —  ^1, 

ioZ>i  =^5+^4  sin  72°+r3  sin  54.°4-^2  sin  36°H-ri  sin  18°, 

10^2 =g2  sin  72°+^i  sin  36°, 

10^3=  —r5+r2  sin  72^+^1  sin  54°  — r4  sin  36°+r3  sin  18°, 

io&4  =  /?i  sin  72°+/j2  sin  36°, 

1065  =  ^1  —  03, 

ioh  =  gi  sin  72°— g2  sin  36°, 

iob7=—r5—r2  sin  72°+ri  sin  54° +^4  sin  36° +^3  sin  18°, 

lobs  =  — /J2  sin  72°+/?!  sin  36°, 

iobQ  =  r5  —  r4:  sin  72°+^3  sin  54°— ^2  sin  36°+ri  sin  18°. 

24-ORDiNATE  Scheme 

yo  yi   y2  ys  y4.  ys   y&  yi  y%  y<i  y\o  y\\  yi2 
y23  y22  y2i  y2o  yi9  yis  yn  yi6  yw  >'i4  yi3 

Sum  Vo      Vi      V2      V3      V4.      V5      Vq      V7      V8      Vg      Z^io    Vn    Vi2 

Difference  wi  W2  1^3  ^4  w^  wq  wi  ws  wq  wio  wn 


86 


EMPIRICAL  FORMULAS 


vo 

Vl 

V2 

V3 

V4 

V5 

1}Q 

^'12        ^'ll          tJlO 

Vo 

Vs 

V7 

Sum 

Po 

pl        p2 

P3 

P^ 

P5 

Pe 

Difference 

qo 

qi      g2 

?3 

q4 

?5 

Wi 

W2 

W3 

W4 

W5 

Wq 

wn 

Wio 

Wo 

w% 

W7 

Sum 

r\ 

r2 

r3 

r4 

rs 

re 

Difference 

S\ 

S2 

S3 

S4. 

^5 

po 

Pi 

p2      p3 

Sl 

S2 

S3 

P6 

^ 

P4 

S5 

S4 

Sum 

k 

/i 

h        h 

Sum 

Tl 

k2 

k3 

Difference 

Mq 

mi 

ni2 

Difference 

m 

W2 

h 

h 

mc 

) 

Wl 

h 
go 

h 
gi 

Sum 

Wi 

> 

Sum 

eo 

ei 

Difference 

ho 

hi 

Difference 

fo 

24ao=go-\-gu 

i2ai=qo-\-^qi-\-W2q3-\-^V;^q2-\-Ciqi+C2q5, 
i2a2  =  mo+im2+iV^mi, 

1.203  =  ^0-^4+1^2(^1-^3-^5), 

i2a4  =  /?o+i^i, 

i2a5  =  qo+C2qi-W3q2-W2qs+iq4:+Ciq5, 

1206  =/o, 

i2a7  =  qo-C2qi  —  i^SQ2+i^q3+iq4—Ciq5, 

i208  =  go-ki) 

I2a9  =  go-?4  +  |V2(-^l+^3+5'5), 

i2aio  =  mo+^m2-iy^Smij 

i2an=qo-Ciqi+^ysq2-W2q3  +  2q4.-C2q5j 
24ai2='ho—hij[ 


DETERMINATION   OF   CONSTANTS 
I2b2  =  ^ki+^V^k2+k3, 

i2b3  =  r2-r6+iV2{ri+r3-r5), 

i2b5  =  Ciri-\-^r2-^V2r3-^Vy4,-\-C2r5+rQ, 

12bQ  =  ki—k3, 

i2b7  =  Ciri  —  ^r2-W2r3+W3^4.+C2r5-rQ, 

I2b8  =  W3(f^l-'^2), 

i2b9  =  rQ-r2+^^2(ri+r3-r5), 

I2bio  =  S3+i(si-\-S5)-W3(^2+S4.), 

12611  =  C2n -1^2+1^2^3 -|V3r44-Cir5 -re, 

^^3  +  1 


87 


Ci 


>.V] 


•96593; 


C2=^^^^  =  .2SS8>2. 

2V  2 

As  an  illustration  let  it  be  required  to  find  a  Fourier   series 
of  24  terms  to  fit  the  data  given  in  the  table  below. 


x" 

y 

x° 

y 

■  x° 

y 

x° 

y 

CX3 

149 

90 

159 

180 

178 

270 

179 

15 

137 

105 

178 

195 

170 

285 

185 

30 

128 

120 

189 

210 

177 

300 

182 

45 

126 

^35 

191 

225 

183 

315 

176 

60 

128 

150 

189 

240 

181 

330 

166 

75 

135 

i6s 

187 

255 

179 

345 

160 

149  137    128    126    128    135    159  178  189  191  189  187  178 

160    166'  176    182    185    179  179  181  183  177  170 

1^0     149297    294    302    310    320    338  357370374366357178 

^1         -23  -38  -50  -54  -50  -20  -I      8      8    12    17 


88  EMPIRICAL  FORMULAS 


149 

297 

294 

302 

310 

320 

338 

178 

357 

366 

374 

370 

357 

A) 

327 

654 

660 

676 

680 

677 

338 

Sto 

-29 

-60 

-72 

-72 

-60 

-37 

-23 

-38 

-50 

-54 

-50 

—  20 

17 

12 

8 

8 

—  I 

fi 

-  6 

-26 

-42 

-46 

-51 

—  20 

Sl 

-40 

-50 

-58 

-62 

-49 

327 

654   660  676 

-40  - 

50  -58 

338. 
665  I 

677  •  680 
[331  1340  676 

ki 

-49  - 

62 

to 

-89  - 

112  -58 

ma 

—  II  - 

-23  - 

20 

m 

9 

12 

665 

1331 

—  II 

-23 

^0 

676 
1341 

1340 
2671 

eo 

—  20 

-31 

-23 

h 

—  II 

-9 

/o 

9 

The  formula  becomes 

;y  =  167. 167  — 19.983  cos  ic  — 3.4 10  cos  2:j;+5.47o  cos  3a; 

—  1.292  cos  4^+. 249  cos  5a;+-75  cos  6ii;+.3io  cos  yx 
+  .458  cos  8x  — .304  cos  9:^;  — .090  cos  lox  — .243  cos  iia; 

—  .083  cos  I2X— 12.779  sin  ic— 16.624  sin  2x  — .323  sin  30; 
+  1.516  sin  40^+1.461  sin  50;  — 2.583  sin  6:j;+.32i  sin  yx 

—  .216  sin  8ic+.676  sin  9X  — .459  sin  iga;  — .639  sin  iia;. 

In  what  precedes  the  period  was  taken  as  27r.  This  is  not 
necessary;  it  may  be  any  multiple  of  27r.  The  process  of  finding 
a  Fourier  series  of  a  limited  number  of  terms  which  represent 
data  whose  period  is  not  2x  will  be  best  set  forth  by  an  example. 
In  the  table  below  the  period  is  7r/3  and  the  values  of  y  are 
given  at  intervals  of  7r/i8.  The  12-ordinate  scheme  can  be 
used  by  first  making  the  substitution 

x=^d  or  d  =  ^x. 


DETERMINATION   OF   CONSTANTS 


x° 

0° 

y 

x° 

0° 

y 

x° 

e° 

y 

oo 

00 

+27.2 

40 

120 

+9-8  ■ 

80 

240 

-"•5 

lO 

30 

-I-34S 

50 

150 

+  8.5 

90 

270 

-17. S 

20 

60 

+  21. 5 

60 

180 

+0.2 

100 

300 

-17.2 

30 

90 

+  10. 1 

70 

210 

-7.1 

no 

330 

+  1. 5 

27.2 

34-5 

21-5 

10 

.  I 

9.8        8.5        0.2 

1-5   - 

-17.2   - 

17 

•5 

-II. 5     -7.1 

^'O 

27.2 

36.0 

4-3     ■ 

-7 

•4 

—  1.7         1.4        0.2 

Wi 

33-0 
27.2 

38.7 
36.0 

27 

.6 

21.3       15.6 
4-3         -7-4 

Po 

0.2 

^•4 

- 

1-7 

27.4 

37-4 

2.6         -7.4 

qo 

27.0 

34-6 

6.0 

33-0 

38.7 

27.6 

27.4        37-4 

15.6 

21.3 

2.6       -7.4 

ri 

48.6 

60.0 

27.6 

lo 

30.0        30.0 

Sl 

17-4 

17-4 

48.6        27.0 
27.6          6.0 

21 .0 


21.0 


The  formula  is 


^'  =  5+9.994  cos  0+8.7  COS  20+3.5  cos  30+.OO6  cos  50 
+  17.31  sin  0+5.023  sin  20+3.5  sin  30  — .01  sin  50. 

Replacing  0  by  its  value  3X, 

^'  =  5+9.994  cos  3X+8.7  cos  6X+3.5  cos  9ii;+.oo6  cos  150; 
+  17.31  sin  3it:+5.o23  sin  6a-+3.5  sin  9X  — .01  sin  150;. 


CHAPTER  VI 

EMPIRICAL   FORMULAS  DEDUCED  BY  THE  METHOD 
OF  LEAST  SQUARES 

In  the  preceding  chapters  we  computed  approximately  the 
values  of  constants  in  empirical  formulas.  The  methods  em- 
ployed were  ahnost  wholly  graphical,  and  although  the  results 
so  obtained  are  satisfactory  for  most  observational  data,  other 
methods  must  be  employed  when  dealing  with  data  of  greater 
precision. 

It  is  not  the  purpose  of  this  chapter  to  develop  the  method 
of  least  squares,  but  only  to  show  how  to  apply  the  method  to 
observation  equations  so  as  to  obtain  the  best  values  of  the 
constants.  For  a  discussion  of  the  subject  recourse  must  be 
had  to  one  of  the  numerous  books  dealing  with  the  method  of 
least  squares.* 

It  was  found  in  Chapter  I  that  the  equation 

y  =  a-\-bx-{-cx^  (i) 

represents  to  a  close  approximation  the  relation  between  the 
values  of  x  and  y  given  by  the  data 


X 

y 

X 

y 

o 

3-I950 

•5 

3.2282 

.1 

3.2299 

.6 

.  3-1807 

.2 

3-2532 

•7 

3.1266 

•3 

3.2611 

.8 

3-0594 

4 

3-2516 

.9 

2.9759 

*  Three  well-known  books  are:  Merriman,  Method  of  Least  Squares; 
Johnson,  Theory  of  Errors  and  Method  of  Least  Squares:  Comstock, 
Method  of  Least  Squares. 

90 


DEDUCED  BY  THE  METHOD  OF  LEAST  SQUARES        91 

where  x  represents  distance  below  the  surface  and  y  represents 
velocity  in  feet  per  second. 

Substituting  the  above  values  of  x  and  yin  (i),  the  following 
ten  linear  observation  equations  are  found : 

a-\- ob-{-  0^  =  3.1950, 
a+. lb +  .01^  =  3.2299, 
(Z+.2&+.04c  =  3.2532, 
«+.3^4-.09^  =  3-26ii, 
(^+.46+. 16^  =  3.2516, 
a+. 56+. 25^=3. 2282, 
a+.6^>+.36(;  =  3.i8o7, 
«+.7^+49^  =  3-i266, 
fl+.86+.64c  =  3.0594, 
fl+.9^>+.8ic  =  2.9759. 

Here  is  presented  the  problem  of  the  solution  of  a  set  of 
simultaneous  equations  in  which  the  number  of  equations  is 
greater  than  the  number  of  unknown  quantities.  Any  set  of 
three  equations  selected  from  the  ten  will  suffice  for  finding 
values  of  the  unknown  quantities.  But  the  values  so  found 
will  not  satisfy  any  of  the  remaining  seven  equations.  Since  all 
of  the  equations  are  entitled  to  an  equal  amount  of  confidence 
it  would  manifestly  be  wrong  to  disregard  or  throw  out  any  one 
of  the  equations.  Any  solution  of  the  above  set  must  include 
each  one  of  the  equations. 

The  problem  is  to  combine  the  ten  equations  so  as  to  obtain 
three  equations  which  will  yield  the  most  probable  values  of 
the  three  unknown  quantities  a,  b,  and  c.  It  is  shown  in  works 
on  the  method  of  least  squares  that  the  first  of  such  a  set  of 
equations  is  obtained  by  multiplying  each  one  of  the  ten  equa- 
tions by  the  coefficient  of  a  in  that  equation  and  adding  the  result- 
ing equations.  The  second  is  obtained  by  multiplying  each  one 
of  the  ten  equations  by  the  coefficient  of  b  in  that  equation 
and  adding  the  equations  so  obtained.  The  third  is  obtained 
by  multiplying  .each  of  the  ten  equations  by  the  coefficient  of 


92  EMPIRICAL   FORMULAS 

c  in  that  equation  and  adding  the  equation  so  obtained.  The 
process  of  computing  the  coefficients  in  the  three  equations  is 
shown  in  the  table.  The  coefficients  oi  a,b,  and  c  are  represented 
by  A,  B,  and  C  respectively,  and  the  right-hand  members  are 
designated  by  N.  The  number  5,  which  stands  for  the  numeri- 
cal sum  oi  A,  B,  C  and  N,  is  introduced  as  a  check  on  the  work. 
It  must  be  remembered  that  this  method  of  finding  the  values 
of  the -constants  holds  only  for  linear  equations. 

The  sum  of  the  numbers  in  the  column  headed  AA=ZAA 
=  io.  The  sum  of  the  numbers  in  the  column  headed  AB  = 
2^45=4.5.  The  sum  of  the  numbers  in  the  column  headed 
-4C  =  2^C  =  2.85.  Also  the  sum  of  the  numbers  in  the  column 
headed  ^iV  =  2^iV"  =  3i.76i6.  These  sums  give  the  coefficients 
in  the  first  equation.*  The  second  and  third  equations  are 
obtained  in  the  same  way. 

The  three  equations  from  which  we  obtain  the  most  probable 
values  of  the  constants  are: 

10  a  -\-  4.56   4-2.85^     =31.7616; 
4.5a    -f2.85^>   -I-2.025C  =14.08957; 
2.85a -f2.o256+i.5333c=  8.828813. 

These  are  called  normal  equations.     From  them  are  obtained 

^=+3-19513; 
b=-\-  .44254; 
^=-   •76531- 
I'he  check  for  the  first  equation  is 

-ZAA+XAB+XAC+ZAN  =  ZAS  =  4g.iii6; 
lor  the  second  equation 

2^5-f255-f  25C -I- 25A^  =  2^5  =  23.46457; 
for  the  third  equation 

2^C-L25C+2CC+2CA^=2C5=i5.237ii3. 
*  Cf.  Wright  and  Hayford,  Adjustment  of  Observations. 


DEDUCED    BY   THE    METHOD    OF    LEAST    SQUARES 


93 


AA 

AB 

ylC 

^iV 

AS 

o 

I 

2 

3 
4 
5 
6 

7 
8 

9 

o 

OI 

04 
09 
16 
25 

36 

49 
64 
81 

3-1950 
3.2299 

3-2532 
3.2611 
32516 
3.2282 
3.1807 
3.1266 
3-0594 
2.9759 

4 
4 
4 
4 
4 
4 

s 

5 
5 
S 

1950 
3399 
4932 
6511 
8116 
9782 
1407 
3166 
4994 
6859 

lO 

4-5 

2.8s 

31.7616 

49. I I 16 

AB 

BB 

BC 

BN 

BS 

0 

0 

0 

0 

.01 

.001 

.32299 

•43399 

04 

.008 

.65064 

.89864 

09 

.027 

•97833 

1-39533 

16 

.064 

1.30064 

1.92464 

25 

.125 

1.61410 

2.48910 

36 

.216 

I . 90842 

3.08442 

49 

•343 

2.18862 

3.72162 

64 

.512 

2.44752 

4-39952 

81 

.729 

2.67831 

5.11731 

4-5 

2.85 

2.025 

14.08957 

23.46457 

AC 

BC 

cc 

CN 

cs 

0 

0 

0 

.0001 

.032299 

•043399 

.0016 

.130128 

.179728 

.0081 

•  293499 

.418599 

.0256 

.520256 

. 769856 

.0625 

. 807050 

I . 244550 

.1296 

I. 14505 2 

1.850652 

.2401 

1.532034 

2.605134 

.4096 

1.958016 

3.519616 

.6561 

2.410479 

4-605579 

2.8s 

2.025 

1-5333 

8.828813 

I5-237II3 

94 


EMPIRICAL   FORMULAS 


The  formula  is 

3'  =  3-i9Si3+-442S4^-.7653i^- 
For  the  purpose  of  comparison  the  observed  values  and  the 
computed  values  are  written  in  the  table,    v  (called  residual) 
stands  for  the  observed  value  minus  the  value  computed  from 
the  formula. 


X 

Observed 

y 

Ccmputed 

y 

V 

»2 

o 

3  1950 

3.1951 

—  .0001 

OOOOOOOI 

.1 

3.2299 

3.2317 

—  .0018 

00000324 

2 

3.2532 

3.2530 

+  .0002 

00000004 

3 

3.2611 

3.2590 

+  .0021 

00000441 

4 

3.2516 

3 .2497 

+  .0019 

00000361 

S 

3.2282 

3.2251 

+  .0031 

00000961 

6 

3.1807 

3.1851 

—  .0044 

00001936 

7 

3.1266 

3.1299 

-.0033 

00001089 

8 

3.0594 

3.0594 

.0000 

00000000 

9 

2.9759 

2.9735 

+  .0024 

00000576 

+  0001  .00005493 

This  method  derives  its  name  from  the  fact  that  the  sum  of 
the  squares  of  the  residuals  is  a  minimum.  A  discussion  of 
this  will  be  found  in  the  books  referred  to  above. 

In  case  the  formula  selected  to  express  the  relation  between 
the  variables  is  not  linear  the  method  of  least  squares  cannot 
be  applied  directly.  In  order  to  apply  the  method  the  formula 
must  be  expanded  by  means  of  Taylor's  Theorem.  Even  when 
the  formula  is  linear  in  the  constants  it  may  be  advantageous 
to  make  use  of  Taylor's  Theorem.  In  order  to  make  this  trans- 
formation clear  we  will  apply  it  to  the  formula  just  considered. 

Suppose  that  there  have  been  found  approximate  values  of 
a,  b,  and  c,  ao,  bo  and  cq,  say,  then  it  is  evident  that  corrections 
must  be  added  in  order  to  obtain  the  most  probable  values  of 
the  constants.    Let  the  corrections  be  represented  by  Aa,  Ab^ 

and  Ac.    And  let 

a  =  ao-\-Aa, 

b  =  bo+Ab, 


c  =  Co-\-Ac. 


DEDUCED  BY  THE  METHOD  OF  LEAST  SQUARES        95 

The  formula  was 

This  may  be  written 

y=f{a,  b,  c)  =f(ao+Aa,  5o+A&,  Co+Ac). 

Expanding  the  right-hand  member 

f(ao+Aa,  bo+Ab,  co+Ac)  =f{ao,bo,Co)+-^Aa+-^Ab-\-^Ai 

9^0         doo         dco 

-{-^^{AaAc)+^^(AbAc)]+   .  .  . 
9^0  9co  9t'o9^o  J 

df 
where   -^  stands  for  the  value  of  the   partial  derivative  of 
doo 

f{a,  b,  c)  with  respect  to  a  and  ao  substituted  for  a,  — ~  stands 

for  the  value  of  the  second  partial  derivative  of  f(a,  b,  c)  with 
respect  to  a  and  ao  substituted  for  a,  etc.  If  ao,  bo,  and  co  have 
been  found  to  a  sufficiently  close  approximation  the  second  and 
higher  powers  of  the  corrections  may  be  neglected. 

^=.; 

9^0 

dbo      ' 

dco 
The  formula  becomes 

y-fiao,  bo,  Co)  =|^Aa+|f  Aj+I^Ac, 
9^0         dOo         dco 

or 

J  —  (flo + box + cox^)  =  Aa + xAb + x~Ac. 


96  EMPIRICAL  FORMULAS 

Selecting  for  the  values  of  ao,  bo,  and  co  those  found  in 
Chapter  I,  the  new  set  of  observation  equations  are 

Aa+ oA6+  oAc=     .0002, 

Aa-\-.iAb-{-.oiAc=  —.0013, 

Aa-\-.2Ab  +  .04Ac=     .0008, 

Aa-\-.^Ab+.ogAc=     .0027, 

Aa-\-.4Ab-\-.i6Ac=     .0024, 

Aa+.sA6+.25Ac=     .0034, 

Aa-\-.6Ab-h-36Ac=  —.0045, 

Aa+.'jAb+.4gAc=  —.0038, 

Aa+.8A6+.64Ac=  —  .0010, 

Aa  +  .gAb+.SiAc=     .0007. 

From  these  are  obtained  the  three  normal  equations 

ioAa+4.5     Ai  +  2.85     Ac=— .0004, 

4.5Aa+2.85  A6+2.025  Ac=— .00203, 

2.85Aa4-2.025A6+i.5333^^= --002059. 
Solving 

Afl= +.00033, 

A5= +.00254, 
Ac  =-.00531, 
which  added  to  the  values  of  ao,  bo,  and  co,  give 

^=  3-19513, 
b=     .44254, 

c= --76531- 
the  same  as  just  found. 

The  above  process  may  be  applied  to  linear  equations  con- 
taining more  than  three  constants.  But  as  the  method  of  pro- 
cedure is  quite  evident  from  the  above  the  general  statement 
of  the  process  will  be  made  with  reference  to  equations  con- 
taining only  three  constants. 


DEDUCED  BY  THE  METHOD  OF  LEAST  SQUARES         97 

Let  the  observation  equations  be  represented  by 

aix+biy-\rc\z  =  ni  pi, 
a2X-\-h2y+C2Z  =  n2  p2, 
a3X+bsy-^C3Z=n3    ps, 


ajnX-\-hmy-\-CmZ  =  nm       pm- 

The  normal  equations  will  then  be 

^pa?  •  X + i:pab '  y + Xpac  •  z  =  Xpan, 
i:pab-x+i:pb^-y+i:pbc'Z  =  7:pbn, 
Zpac'X+Xpbc-y-h^pc^  'Z  =  '2pcn, 

where  a,  b,  c,  and  n  are  observed  quantities,  and  x,  y,  and  z 
are  to  be  determined,  pi,  p2,  ps  .  .  .  pmSLve  the  weights  assigned 
to  the  observation  equations.  In  the  problem  treated  at  the 
beginning  of  the  chapter  the  weight  of  each  equation  was  taken 
as  unity. 

It  was  stated  on  a  preceding  page  that  when  a  formula  to  be 
fitted  to  a  set  of  observations  is  not  linear  in  the  constants  it 
must  be  expanded  by  Taylor's  Theorem. 

Take  as  an  illustration  a  problem  considered  in  Chapter  IV. 

The  formula  considered  was 


y 

=fiA,B, 

m,n) 

=Ax"'+Bx'', 

a/. 

=x^% 

9/. 
dBo 

=x^\ 

9/. 
9wo 

=  Aox'^^ 

'  log  X, 

a/. 

dno 

=  Box^'^ 

logx; 

EMPIRICAL   FORMULAS 


y  =f(A ,  B,  m,  n)      =/Uo,  Bo,  m,  «o)  +^Ayl  +^^B 

Swo         dm 
y-f{Ao,  Bo,  mo,  no)  =:^A^  +^a5+|^Aw+|^Aw. 

The  observation  equations  will  be  of  the  form 

^-^A^\-^^B^-^^m+^^n=y-yo. 
dAo  dBo  dmo  dm 

Assume  the  approximate  values  found  in  Chapter  IV. 
A=  1.522, 
B=-.6Ss, 
w=     .55, 
n=   1.4. 


.55 


X-   

Ao 

Bo 

logac 

Atix^^logx. 
5o«"°  log  X. 


X-    

log  a; .  .. 

A^^'^Xogx. 
Box'"'  log  X. 


05 

.10 

•15 

.20 

•25 

19 

.28 

•35 

•  41 

•47 

02 

.04 

.07 

.10 

•14 

I 

522 

— 

685 

—  2 

996 

-2.303 

-1.897 

—  1 . 609 

-1.386 

— 

88 

-  -99 

—  1.02 

—  1. 01 

-  .98 

03 

.06 

.09 

.12 

•  14 

The  new  observation  equations  become 

.igAA-\-.02AB—  .SSAm+.o^An=  .0004, 
.28Ai4+.04A5—  .ggAm-\-.o6An=  .0002, 
.35A^  +  .o7A^  — i.02Aw+.09An=  —.0001, 


•  30 

•  52 
.19 


—  1 . 204 

-  -94 
•15 


•35 

.40 

•45 

•50 

.56 

.60 

•  64 

.68 

•23 

.28 

•33 

.38 

-1.050 

—  0.916 

-0.799 

-0.693 

—  0. 

-  .90 

-  .84 

-  .78 

-  .72 

-  . 

.16 

•17 

.18 

.18 

55 
72 

43 
598 
66 
18 


DEDUCED  BY  THE  METHOD  OF  LEAST  SQUARES        99 

.4iAA+.ioAB  —  i.oiAm-\-.i2An=  .ocxx), 
.47A^-f  .14AJ5—  .98AwH-.i4A;z=  .0013, 
.52A^+.i9AB—  .94Aw+.i5A;z=  —  .0001, 
.S6AA-\-.2^AB—  .goAfn-\-.i6An=—.ooigf 
.6oA^  +  .28A5—  .84Aw4-.i7A;z=  —  .0016, 
.64A^+.33A5—  .78Aw  +  .i8Aw=  —  .0001, 
.68A^  +  .38A5—  .72Aw+.i8A;^= -.0001. 
.72A^+.43A5—  .66Am-\-.iSAn=     .0011. 

From  these  the  four  normal  equations  are  obtained 

2.96oA^4-i.32iA5— 4.637AW+  .8o6Aw=  —.00071, 

i.32iA^+   .642A5  — i.8o2Aw4-  .359A;?= —.00031, 

— 4.637A^  — 1.802A5+8.737AW  — i.253A7^  =  +.ooo85, 

.8o6A^+  .359A5  — 1.253AW+  .22iAw= —.00023. 

From  which 

A^  =  -.oo68, 

AJ5  =  4-.oii2, 
Aw  =—.0022, 
Aw  =  —  .0070. 

These  corrections  being  applied  the  final  formula  becomes 
>;  =  1.5 1 5 2a;*^478_  6^^3^1.393^ 


CHAPTER  VII 

INTERPOLATION.— DIFFERENTIATION  OF  TABULATED 
FUNCTIONS 

Interpolation  ^  ' 

In  Chapter  II  we  found  that  the  formula 

XL  y= 

.025+.2525:*;+2.5ac2 

represents  to  a  fair  degree  of  approximation  the  values  of  y 
given  by  the  data.  Any  other  value  of  y^  within  the  range  of 
values  given,  can  be  obtained  in  the  same  way.  This  rests  on 
the  assumption  that  the  formula  derived  expresses  the  law  con- 
necting X  and  y.  For  example,  the  value  of  y  corresponding 
to  a:  =  1.05  will  be 

^".025-}-.2525(i.05)  +  2.5(i.05)2~°-^45. 

When  a  formula  is  used  for  the  purpose  of  obtaining  values 
of  y,  within  the  range  of  the  data  given  it  is  called  an  inter- 
polation formula.  Interpolation  denotes  the  process  of  calcu- 
lating under  some  assumed  law,  any  term  of  a  series  from  values 
of  any  other  terms  supposed  given.*  It  is  evident  that  empirical 
formulas  cannot  safely  be  used  for  obtaining  values  outside 
of  the  range  of  the  data  from  which  they  were  derived. 

*  For  a  more  extended  discussion  of  the  subject  the  reader  is  referred 
to  Text -book  of  the  Institute  of  Actuaries,  part  II  (ist  ed.  1887,  2nd  ed. 
1902),  p.  434;  Encyklopadie  der  Mathematischen  Wissenchaften,  Vol.  I, 
pp.  799-820;  Encyclopedia  Britannica;  T.  N.  Thiele,  Interpolationsrechnung. 

As  to  relative  accuracy  of  different  formulas,  see  Proceedings  London 
Mathematical  Society  (2)  Vol.  IV.,  p.  320. 

100 


INTERPOLATION  '  3  '  -  -  '  '       '     '    '  JQJ 

There  are  two  convenient  formulas  for  interpolation  which 
will  be  developed.* 

The  first  one  of  these  requires  the  expression  for  yx+n  in 
terms  of  yx  and  its  successive  differences,  yx  represents  the 
value  of  a  function  of  x  for  any  chosen  value  of  x,  and  yx+n 
represents  the  value  of  that  function  when  x+n  has  been  sub- 
stituted for  X. 

yx+i=yx+^yx; 
yx+2=yx-{-Ayx+A(yx+^yx) 

=yx+2Ayx+A^yx; 
yx+  3=yx+2Ayx-\r  ^yx  -\-A{yx-\-  2Ayx + A^y^ 

=yx-\-2>^yx^-^A^yx-{-A^yx\ 
yx+A=yx-\-2>^yx-\r^A^yx-\-A^yx-\-A{yx+2>^yx-\-2>^^yx+A^yx) 

=yx-\-A^yx-\-6A^yx-\-A^^yx-{-A'^yx- 

These  results  suggest,  by  their  resemblance  to  the  binomial 
expression,  the  general  formula 

,     ^      ,n{n  —  i)^c.     .  n{n  —  i){n  —  2)  ^^     .    ^ 
yx+n=yx-\-nAyx-]-^ -^^yx^ — ^ r^ ^A33;^+etc. 

If  we  suppose  this  theorem  true  for  a  particular  value  of  w, 
then  for  the  next  greater  value  we  have 

,     .      .n(n  —  i)^^     .  n(n  —  i)(n—2)  .^     ,    . 
yx+n+i=yx+nAyx-{ -, A^yx-\ — ^ p ^A^^y^^+etc., 

+Ayx+nA^yx+—, — ^A^^y^+etc, 

=  yx+{n  +  i)Ayx-i-^     I        A^yx+- 7-^ ^A3;y:,+etc. 

The  form  of  the  last  result  shows  that  the  theorem  remains 
true  for  the  next  greater  value  of  n,  and  therefore  for  the  next 

*  See  Chapter  III,  Boole's  Finite  Differences. 


102 


EMPIRICAL  FORMULAS 


greater  value.     But  it  is  true  when  w  =  4,  therefore  it  is  true 
when  w  =  5.    Since  it  is  true  for  w  =  5  it  is  true  when  «  =  6,  etc. 
If  now  o  is  substituted  for  x  and  x  iov  n,  it  follows  that 

,       ^         ,x(x—l)^^        ,  x{X'-l){x  —  2)  ^^        ,      . 
|2  13 

If  A**^'.  =0,  the  right-hand  member  of  the  above  equation 
is  a  rational  integral  function  of  x  of  degree  n  —  i.  The  formula 
becomes 

,       ^         ,x{x—l)^^        ,  x{x—l)(x  —  2)  ^^ 

yx=yo-\-xAyo+-^. — -A^yo-\-— f^ -A^yo+  .  . 


13 

^x{x-l){x-2)    .    ■    .    (^-^-^2)^n-l 

\n  —  i 


(i) 


Formula  (i)  will  now  be  applied  to  problems.  //  must  not 
be  forgotten  that  in  applying  this  formula  x  is  taken  to  represent 
the  distance  of  the  term  required  from  the  first  term  in  the  series, 
the  common  distance  of  the  terms  given  being  taken  as  unity. 

I.  Required  to  find  the  value  of  y  corresponding  to  a;  =  .4 2 5 
having  given  the  values  under  XIX.  In  the  interpolation 
formula  x  =  . 5. 

yo  y\  yi  yz 


•730 

•1S1 

.780 

AyQ 

.027 

.023 

.020 

A^^^o.... 

—  .004 

-.003 

l^y^ 

.001 

.800 


y=yo^-hAyQ  -  {A^yo^-^A^yo 

=  .730+ .0135 +  .0005 +  .0001 

=  .744. 

This  is  the  same  as  given  by  XIX. 

2.  Find  the  value  of  y  corresponding  to  ^  =  2.3.     :*;  in  the 
formula  will  have  the  value  f  if  we  take  >'o=— .1826  when 

X  =  2. 


INTERPOLATION 


103 


yo 

yi 

y2 

,  y^ 

y^ 

y5 

-.1826  - 

4463 

-•7039 

-.9582 

— 1.2119 

-1.4677 

-.2637  - 

2576 

-•2543 

-•2537 

-  .2558 

.0061 

0033 

.0006 

—.0021 

-.0028  - 

0027 

—  .0027 

.CX30I 

.0000 

—  .0001 

Ayo 

A^yo 
AryQ 


-.i826+f(-.2637)  +  i^^Coo6i)  +  i^-4^-^ 


(-.0028) 


K-f)(-«(-¥) 
24 


(.0001) 


-•3417- 


3.  The  following  example  is  taken  from  Boole's  Finite  Differ- 
ences. Given  log  3.14  =  .4969296,  log  3.15  =  .4983106,  log  3.16  = 
.4996871,  log  3.17  =  .5010593;  required  an  approximate  value  of 
log  3-I4I59- 

>'o  yi  y2  73 


AjQ. 

d?yQ 


.4969296 

.4983106 

.4996871 

•5010593 

.0013810 

.0013765 

.0013722 

— . 0000045 

-.0000043 

. 0000002 

Here  the  value  of  x  in  the  formula  is  equal  to  0.159. 
3;;,  =  . 4969296  +  (.i59)(-ooi38io)+  (-.0000045) 


,    •l59(-l59-l)Cl59-2)^  r.r.r.r^^.\ 

i (.0000002 ) 

6 


=  .4971495. 


104  EMPIRICAL  FORMULAS 

This  is  correct  to  the  last  decimal  place.  If  only  two  terms 
had  been  used  in  the  right-hand  member  of  the  formula,  which 
is  equivalent  to  the  rule  of  proportional  parts,  there  would 
have  been  an  error  of  3  in  the  last  decimal  place.  The  rapid 
decrease  in  the  value  of  the  differences  enables  us  to  judge 
quite  well  of  the  accuracy  of  the  results.  The  above  formula 
can  be  appUed  only  when  the  values  of  x  form  an  arithmetical 
series. 

In  case  the  series  of  values  given  are  not  equidistant,  that  is, 
the  values  of  the  independent  variable  do  not  form  an  arithmetical 
series,  it  becomes  necessary  to  apply  another  formula. 

Let  >,  >,  >'c,  yd,  .  .  .  yt  be  the  given  values  corresponding 
to  a,  h,  c,  d,  .  .  .  k  respectively  as  values  of  x.  It  is  required 
to  find  an  approximate  expression  for  y^,  an  unknown  term 
corresponding  to  a  value  of  x  between  x^a  and  x  =  k. 

Since  there  are  n  conditions  to  be  satisfied  the  expression 
which  is  to  represent  all  of  the  values  must  contain  n  constants. 
Assume  as  the  general  expression 

y,=A-\-Bx+Cx^+Do^-^  .  .  .  ■\-Nx''-\ 

Geometrically  this  is  equivalent  to  drawing  through  the  n 
points  represented  by  the  n  sets  of  corresponding  values  a 
parabola  of  degree  «  — i. 

Substituting  the  sets  of  values  given  by  the  data  in  the 
equation  above  n  equations  are  obtained  from  which  to  determine 
the  values  oi  A,  B,  C,  etc., 

ya=A+Ba+Ca^+Da^-{-  .  .  .  iVa"-^; 
yi,=A-\-Bb+Cb^+DP+  .  .  .  iV^""^; 


yt=A+Bk+Ck^+Dk^+  .  .  .  Nk^'-K 

But  the  solution  of  these  equations  would  require  a  great 
deal  of  work  which  can  be  avoided  by  using  another  but  equiva- 
lent form  of  equation. 


INTERPOLATION 

Let y:,  =  A(x-b)(x-c)(x-d)  .  . 

.  (x-k) 

+B(x-a)(x-c)(x-d)    .  . 

.  (x-k) 

+C{x-a)(x-b)(x-d)  .  . 

.  (x-k) 

+D{x-a)(x-b)ix-c)  .  . 

.  (x-k) 

+  etc.  to  n  terms. 

105 


Each  one  of  the  n  terms  on  the  right-hand  side  of  the  equation 
lacks  one  of  the  factors  x  —  a,  x  —  b,  x—c,  x—d,  .  .  .  x—k, 
and  each  is  affected  with  an  arbitrary  constant.  The  expression 
on  the  right-hand  side  of  the  equation  is  a  rational  integral 
function  of  x. 

Letting  x  =  a  gives 


and 


ya  =  A{a  —  b)(a—c)(a—d)  .  .  .  a—k, 


ya 


{a  —  h){a  —  c){a—d)  .  .  .  a  —  k' 
Letting  x  =  h  gives 
B  = 


y^ 


{h-a){h-c){h-d)  .  .  .  {h-k)' 

Proceeding  in  the  same  way  we  obtain  values  for  all  of  the 
constants  and,  finally, 

{x  —  h){x  —  c){x  —  d)  .  .  .  {x  —  k) 


yx=ya 


+yi 


{a  —  h){a  —  c){a  —  d) 
{x  —  a){x  —  c){x  —  d) 


+yc 


{h-a){h-c){b-d)  . 
{x  —  a){x  —  h){x  —  d)  . 


+yd 


{c—a){c—h){c—d)  . 
(x  —  a){x  —  b)(x  —  c)  . 


{d-a){d-h){d-c)  . 


+yt7 


{x  —  a)(x  —  b)(x  —  c) 


(k-a){k-b){k-c)  . 


.  (a-k) 
.  (x  —  k) 


.  (b-k) 

.  (x-k) 


.  (c-k) 

.  {x  —  k) 


.  {d-k) 


(2) 


106  EMPIRICAL   FORMULAS 

This  is  called  Lagrange's  theorem  for  interpolation. 

I.  Apply  formula  (2)  to  the  data  given  under  formula  XIX 
for  finding  the  value  of  y  corresponding  to  ic  =  0.425.  Select 
two  values  on  either  side  of  the  value  required, 

<^  =  -35>  >'a  =  .695, 
6  =  .40,  >  =  .730, 
^  =  •45,  yc  =  '1Sh 
d=.so,  yd=. 7S0. 

X  in  the  formula  must  be  taken  as  0.5. 


(-i)(-2)(-3)     ^  '-^  '(i)(-i)(-2) 

=.744. 

2.  Required  an  approximate  value  of  log  212  from  the  fol- 
lowing data: 

log  210  =  2.3222193, 

log  211  =  2.3242825, 

log  213  =  2.3283796, 

log  2 14  =  2.3304138. 

=  2.326359. 

This  is  correct  to  the  last  figure. 

In  case  the  values  given  are  periodic  it  is  better  to  use  a 
formula  involving  circular  functions.  In  Chapter  V  the  approxi- 
mate values  of  the  constants  in  formula  XX  were  derived.  This 
formula  could  be  used  as  an  interpolation  formula.  But  on 
account  of  the  work  involved  in  determining  the  constants  it  is 


INTERPOLATION 


107 


much  more  convenient  to  use  an  equivalent  one  which  does  not 
necessitate  the  determination  of  constants.*  The  equivalent 
formula  given  by  Gauss  is 


sin  \{%  —  }))  sin  \{x—c) 


sin  \{a  —  }))  sin  \{a  —  c) 
sin  \{x  —  (i)  sin  \{x  —  c) 


sin  1(6  — a)  sin  \i})  —  c) 
sin  \{x—a)  sin  \{x  —  })) 


sin  \{c  —  o)  sin  \{c  —  })) 
+  etc 


.  sin  \{x—'li\ 


.  sin  \{a  —  )i) 
.  sin  \{x—'li) 


.  sin  \i})  —  K) 
.  sin^(x  — ^) 


.  sin|(c  — ^) 


(3) 


It  is  evident  that  the  value  of  ya  is  obtained  from  this  formula 
by  putting  x  =  a.  The  value  of  y^  is  obtained  by  putting  x  =  h^ 
and  yc  by  putting  %  =  c. 

The  proof  that  (3)  is  equivalent  to  XX  need  not  be  given 
here. 

Let  it  be  required  to  find  an  approximate  value  of  y  cor- 
responding to  x  =  42°  from  the  values  given. 


From  (3) 
3'  =  (io.i) 


X 

y 

30° 

10. 1 

40" 

9.8 

50" 

8.5 

sin  i°sin  (^4°) 
sin  (  —  5°)  sin(  — 10°) 


■K9-8) 


+  (8.5) 


sin  6°  sin  (—4°) 
sin  s°  sin  (-5°) 

•        ^  O      •  O 

sm  6   sin  i 
sin  10°  sin  5° 


-     rxn  T^i:^II5)C^698)     ,     .  (.1045) (.0698) 

(.1045)  (.01 75) 

(.i736)(.o872) 
=  9.618. 


+  (8.5) 


*  Trigometrische    Interpolation,    Encyklopadie    der    Mathematischen 
Wissenchaften,  Vol.  II,  pt.  I,  pp.  642-693. 


108  EMPIRICAL  FORMULAS 

A  better  result  would  have  been  obtained  by  using  four  sets 
of  values. 

Differentiation  of  Tabulated  Functions 

It  is  frequently  desirable  to  obtain  the  first  and  second 
derivatives  of  a  tabulated  function  to  a  closer  approximation 
than  graphical  methods  will  yield.  For  that  purpose  we  will 
derive  differentiation  formulas  from  (i)  and  (2).    From 

,       .         ,  x(x—l).o        .  x(x—l)(x  —  2).^ 

^x{x-i){x-2)(x-s)^^^^^^ 

k 


By  differentiating  it  follows  that 

,      ^      ,  2ic— i.o      ,  3aj2-6:r+2., 
yx=Ayo+-Y-^y(i-^ j ^3^0 

^4^-i2.H-...-6^,^^^ (^^ 

Differentiating  again 
yx"=A^yo+(x-i)A^yo+(hx^-x+H)A'yo+  .....    .     .     (s) 

As  an  illustration  let  it  be  required  to  find  the  first  and 
second  derivatives  of  the  function  given  in  the  table  below  and 
determine  whether  the  series  of  observations  is  periodic* 

The  consecutive  daily  observations  of  a  function  being 
0.099833,  0.208460,  0.314566,  0.416871,  0.514136,  0.605186, 
0.688921,  0.764329,  show  that  the  function  is  periodic  and  deter- 
mine its  period. 

*  Interpolation  and  Numerical  Integration,  by  David  Gibb. 


INTERPOLATION 


109 


From  the  given  observations  the  following  table  may  be 
ttpn ! 


written: 


y=f(x) 

0-099833 
0.208460 
0.314566 
0.416871 
0.514136 
0.605186 
0.688921 
0.764329 


0.108627 
0.106106 
0.102305 
(-)o.097265 
(-)o.o9io5o 
(-)o.o83735 
(-)o.o754o8 


From  (4) 

y\=     .108627 
.001260 


. 109887 
.000437 

. 109450 


— .000427 
— .000010 

-.000437 


73=   .102305  —.000392 
.002520  —.000019 


. 104825 
.000411 

. 104414 


.000411 


.002521 
.003801 
.005040 
.006215 

•007315 
.008327 


^2 


—  .001280 
(+) -.001239 

(+)-.ooii75 
(+)  — .001100 
(+)  — .001012 


A< 


.000041 
.000064 
.000075 
.000088 


106106 
001900 


108006 
000429 


107577 

097265 
003108 


100373 
000389 


— .000413 
— .000016 

— .000429 


— .000367 
— .000022 

— .000389 


099984 


For  the  remaining  first  derivatives  the  order  must  be  reversed 
and  the  resulting  sign  changed. 

^5=- .097265         .002520         ^6=  — .091050 
,000010         .000413  —.000016 


.003108 
.000392 


097275 
002933 


002933 


094342 


— .091066 
.003500 

.087566 


.003500 


110 


EMPIRICAL  FORMULAS 


/7=-.  083735 
,000019 


083754 
004025 


,003658 
.000367 

004025 


— .006279 
.000038 

— .006241 

y'7=-. 007315 
— .001100 


.008415 
.000069 

,  008346 


y8=-.o754o8 
— .000022 


.079729 

From  (5) 

y'i  = -.002521 

.001280 

y"2 

.001318 

.000038 
.001318 

— .001203 

/';,=  -.  005040 

.001175 

y'4 

.001244 

.000069 
.001244 

-.003796 

y'5=-.  005040 

A 

-.001239 

075430 

.004501 
.070929 


003801 
001298 


A." 

y  8 


002503 

006215 
001181 


005034 

006215 
001175 


007390 
000059 


007331 

008327 
001012 


009339 
000081 


009258 


.004164 
000337 

004501 


.001239 
,000059 

001298 

OOIIOO 

,000081 
,001181 


X 

y 

y' 

y" 

y 

I 

.099833 

. 109450 

— .001203 

—  .0121 

2 

. 208460 

.107577 

— .002503 

— .0120 

3 

.314566 

. 1044 I 4 

-.003796 

— .0121 

4 

.416871 

.099984 

-.005034 

— .0121 

INTERPOLATION 

111 

X 

y 

y 

y" 

y 

5 

.514136 

.094342 

.006241 

—  .0121 

6 

.605186 

.087566 

.007331 

— .0121 

7 

.688921 

.079729 

.008346 

— .0121 

8 

.764329 

.070929        — 

.009258 

— .0121 

Jf 


Since  —  is  very  nearly  constant  and  equal  to  —.0121,  the 

y 

corresponding  differential  equation  is 

y''-\-.oi2iy  =  o, 
whose  solution  is  •  . 

3'  =  ^  coso.iix+^sino.iix. 
This  shows  that  y  is  a,  period  function  of  x,  and  its  period  is 

2ir  J 

,  or  57.12  days. 

o.ii 

Convenient  formulas  for  the  first  and  second  derivatives  may 
also  be  obtained  by  differentiating  Lagrange's  formula  for  inter- 
polation.    Using  five  points  the  formula  is 

_      (x  —  b)(x  —  c)(x—d){x  —  e)         (x  —  a)(x—c)(x  —  d)(x—e) 
^"~^"  {a-b)(a-c)(a-d)(a-ey^'  {b-a){h-c){b-d){h-e) 

{x  —  a){x  —  h){x  —  d){x  —  e)         {x  —  a){x  —  h){x  —  c){x  —  e) 
'^^^  {c-a){c-h){c-d){c-e)  "^^'  {d-a){d-h){d-c){d-e) 

{x  —  a){x  —  h){x  —  c){x  —  d)  .  . 

'^^^  ie-a){e-b){e-c){e-d) ^^^ 

Selecting  the  points  at  equal  intervals  and  letting 

e—d  =  d—c  =  c  —  b  =  b  —  a  =  hj 

and  differentiation 

ya=-\l-2sya+4^yb-3^yc-\-i6yd-3ye], 
y'b  =  — rl-  33'a-io>+i8>'c-   6yd  +  ye], 

12^ 


112 


EMPIRICAL  FORMULAS 


^  =  77l[         y^-  ^>  +  Sjd-yJ, 


iih 


y'^^—X-     >'a+  6j6-i83;c+io>'d+33'J, 


Differentiating  again 


y'«  =  7;^t^5>'a - 1043'*+ 1 ^Ayc -  56>'d+ 1 !>], 


^"*"72A2'"^"~  ^°^'"^     ^^'^"^     43'd-3'J, 
y'c  =  ^[-3'a+   i6>-  30^0+   i6>'d->'e], 
yd=— pl-^^     4>+     6>'c-  2o>'d+ii>], 
ye  =  -^[ii3'a-  56>+ii4>;c-io4>'d+353'J- 


The  results  of  applying  these  formulas  to  the  function  given 
are  expressed  in  the  table  below. 


X 

y 

/ 

y 

I 

099833 

. IO945I 

.001203 

2 

208460 

•  107583 

002524 

3 

314566 

.104415         - 

003804 

4 

416871 

.099986         — 

005045 

5 

514136 

•094347 

006221 

6 

605186 

.087568         - 

007322 

7 

688921 

•079733 

008334 

8 

1j  _ 

764329 

r_  ?  1 

.070929 
11   'j 1  J 1 

009258 
1    -1 

These  results  agree  fairly  well  with  those  previously  obtained. 
It  is  probable  that  the  formulas  derived  from  the  interpolation 
formula  give  the  most  satisfactory  results. 


INTERPOLATION 


113 


As  another  application  let  us  find  the  maximum  or  minimum 
value  of  a  function  having  given  three  values  near  the  critical 
point. 

Let  ya^  jb,  and  yc  be  three  values  of  a  function  of  x   near 
its  maximum  or  minimum  corresponding  to  the  values  of  Xy 
a,  h,  and  c  respectively. 
From  (2) 

^      {x-h){x-c)        {x-d){x-c)         {x-a){x-h) 
^'    ^"  {a-h){a-cy^\h-a){h-cy^'  {c-a){c-by 

Equating  to  zero  the  first  derivative  with  respect  to  x 

2x  —  a  —  b 


2X  —  b  —  c 

y<'  7 — 1X7 — ^+3^* 


2X  —  a- 


Vyc 


{a-b){a-c)  '  ■""  {b-a){b-c)  '  '   {c-a){c-b) 
_  ya(b^  -  c^)  -i-y,(c^  -  a^)  +ycia^  -  b^) 


=  0; 


(6) 


2[ya{b-c)-\-yi,{c-a)-\-yc(a-b)] 

This  is  equivalent  to  drawing  the  parabola 

y=A+Bx-\-Cx^ 

through   the  three  points  and  determining  its  maximum  or 
minimum. 

From  the  table  of  values 


6.0 

6.5 

7.0 


10.05 
10.14 
10.10 


the  abscissa  of  the  maximum  point  is  found  from  (6) . 

(io.o5)(-6.75)  +  (io.i4)(i3)  +  (io.io)(-6.25)^ 
2[(io.o5)(-.5)  +  (io.i4)(i)  +  (io.io)(-.5)        ^'^^^ 


>;  =  10.1424. 


CHAPTER  VIII 
NUMERICAL  INTEGRATION 

Areas 

An  area  bounded  by  the  curve,  y=f{x),  the  axis  of  x,  and 
two  given  ordinates  is  represented  by  the  definite  integral 


=r 


ydx, 

where  the  ordinates  are  taken  at  it;  =  a  and  x  =  n.  It  may  be 
said  that  the  definite  integral  represents  the  area  under  the 
curve,  or  that  the  area  under  the  curve  represents  the  value  of 
the  definite  integral. 

If  a  function  is  given  by  its  graph,  it  is  possible,  by  means 
of  the  planimeter,  to  find  roughly  the  area  bounded  by  the  curve, 
two  given  ordinates  and  the  x  — axis,  or,  what  amounts  to  the 
same  thing,  the  area  enclosed  by  a  curve.  This  method  is  used 
in  finding  the  area  of  the  indicator  diagrams  of  steam,  gas  or 
oil  engines,  and  various  other  diagrams.  The  approximations 
in  these  cases  are  close  enough  to  satisfy  the  requirements. 

If,  however,  considerable  accuracy  is  sought,  or  whenever 
the  function  is  defined  by  a  table  of  numerical  values  another 
method  must  be  employed. 

Mechanical  Quadrature  or  Numerical  Integration  is  the  method 
of  evaluating  the  definite  integral  of  a  function  when  the  func- 
tion is  given  by  a  series  of  numerical  values.  Even  when  the 
function  is  defined  by  an  analytical  expression  but  which  can- 
not be  integrated  in  terms  of  known  functions  by  the  method 
of  the  integral  calculus,  numerical  integration  must  be  resorted 
to  for  its  evaluation. 

The  formulas  employed  in  numerical  integration  are  derived 
from  those  established  for  interpolation. 

114 


NUMERICAL   INTEGRATION  115 

In  interpolation  it  was  found  that  the  order  of  differences 
which  must  be  taken  into  account  depends  upon  the  rapidity 
with  which  the  differences  decrease  as  the  order  increases. 
This  is  also  true  of  numerical  integration.  It  is  the  same  as 
saying  that  if  the  series  employed  does  not  converge  the  process 
will  give  unsatisfactory  results.  An  illustration  will  be  given 
later. 

Formulas  for  numerical  integration  will  be  derived  from  (i) 
of  Chapter  VII. 

In  this  formula  it  was  assumed  that  the  ordinates  are  given 
at  equal  intervals. 

,      ^         .x{x—l).^        ,  x{x—l){x  —  2)  .^ 

yx=yo+xAyo+^ -A^yo+      — , ^A^yo 

^  x{x-i)(x-2)(x-s)^.      I  x{x-i)(x-2)(x-s){x-4)^.^ 

^  x{x-i){x-2)(x-s)(x-4){x-s)^,    j ^^-^ 

[6  ....... 

Integrating  the  right-hand  member, 
\  ydx=yQ    I    dx-\-^yo  \   xdx-\ — r^  I   x{x  —  i)dx 

0  Jo  Jo  \2_J 

+^Vx{x-l){x-2)dx 

13  Jo 

k  Jo 

+~j^  I   x(x-i){x-2){x-^){x-4){x-s)dx+  .  .  . 
2  \3       2/    |2       V4  /    |3 


_^/^_3fi4_^lrf_        \A4y, 
\5       2         3  /   k 


116  EMPIRICAL  FORMULAS 

\6  4  3  /Is 

\7      2  '  4  3  /    P 

The  data  given  in  any  particular  problem  will  enable  us 
to  compute  the  successive  differences  of  yo  up  to  A"yo.  On  the 
assumption  that  all  succeeding  differences  are  so  small  as  to 
be  negligible  the  above  formula  gives  an  approximate  value 
of  the  integral.  It  is  only  necessary  to  assign  particular  values 
tow. 

Let  «  =  2,  then 

yxdx  =  2yo + 2  Ayo +iA2yo, 

Ayo  =  yi-yo, 

A^yo=Ayi  -Ayo=y2-yi  -yi+yoy 
=y2-2yi+yo. 

Substituting  these  values  in  the  above  integral  it  becomes 
I   yJx  =  2yo-\-2yi-2yo+iy2-hi+^yo, 

_^yo+4yi-\-y2 

3 

This  is  equivalent  to  assuming  that  the  curve  coincides  with 
a  parabola  of  the  second  degree. 

If  the  common  distance  between  the  ordinates  is  h,  the 
value  becomes 

'2h 

ydx=U(yo+4yi-^y2) (7) 


X 


n»=3 

ydx=syo+^^yo+i^^yo+iA^yo, 

Ayo=yi-yo, 
A^yo  =  Ayi-Ayo=:y2-  2yi  +yo, 


i 


NUMERICAL  INTEGRATION  117 

A^^'o  =  A^>'i  —  ^^yo  =  ^y2  —  Ayi  —  Ayi + Ayo 

=y3-3y2+3yi-yo- 

Substituting  these  values  in  the  equations, 

1  ydx  =  syo+^yi-ho+iy2-hi+ho-\-iy3-h2+hi-h'o, 

=iyo+iyi+h2+iys, 
=i(yo+3yi+3y2+y3)> 

If  the  common  distance  between   the  ordinates  is  h  the 
formula  becomes 

•3ft 


x 


=ih(yo+3yi+3y2+y3) (8) 


This  is  equivalent  to  assuming  that  the  curve  coincides  with 
a  parabola  of  the  third  degree. 

If  there  are  five  equidistant  ordinates,  h  representing  the 
distance  between  successive  ordinates 


X 


^'^■j^  -  i4(yo+y4)  +  64(yi  +y^)  +  2^y2j^^    ...  (9) 


If  the  area  is  divided  into  six  parts  bounded  by  seven  equi- 
distant ordinates  the  integral  becomes 


X 


6 

ydx  =  6>'o  4- 1 8  A>'o  +  2  7  A^^yo  +  24A^yQ  +  ^A^yo 
+HA^>'o+i^A6:Vo. 


Sine  3  the  last  coefficient,  1^,  differs  but  slightly  from  A 
and  by  the  assumption  that  A^yo  is  small  the  error  will  be  slight 
if  the  last  coefficient  is  replaced  by  1^. 

Doing  this  and  replacing 

Ayohy  yi-yo, 
A^yQhYy2-2yi+yo, 
A^yo  by  yz-:^y2+:^yi  -yo, 


118 


EMPIRICAL   FORMULAS 


A^^'o  by  y4-4ys+6y2  -4>'i+3'o, 
A^^o  by  ys  -  53'4  +  loya  - 1 oyo  +  syi  -  >'o, 
A<^yo  by  /  -  6y5 -f  1 53'4  -  soys  + 1 53'2  -  6y i  +yo, 
gives  the  formula 


X 


ydx  =  Ty'b'o+y2+y4+y6+5(>'i  +y5)  +6y3].    .    (lo) 


The  application  of  these 
formulas  is  illustrated  by 
finding  the  area  in  Fig.  27. 


Fig.  27. 


By  (7) 

A  =4/j(yo+4>'i  +  2y2H-4y3+2y4+43'5+2y6+4y7+2y8+43'9 

+  2yio+4yii+yi2). 
By  (8) 
-4=|/f(yo+33'i+33'2+2y3+3>'4+33'5+2y6+3>'7+3y8+2y9 

+3yio-\-3yn+yi2)' 

By  (9) 

-4=AMi4(jo+2y4  +  2y8+yi2)+64(yi+y3+y6+3'7+y9+yii) 

+24(y2+y6+yio)]. 
By  (10) 

^=A%o+y2+y4+2y6+y8+yio+yi2+5(3'i+>'5+>'7+yii) 
+6(y3+y9)]..  . 

I.  A  rough  comparison  of  the  approximations  by  the  use 
of  these  formulas  will  be  obtained  by  finding  the  value  of 

— .    The  value  of  this  definite  integral  is  log  13  =  2.565.     It 

1     X 

is  also  equal  to  the  area  under  the  curve 


y= 


NUMERICAL   INTEGRATION     .  119 

from  ^  =  1  to  it; -=13.  Dividing  the  area  up  into  12  strips  of 
unit  width  by  13  ordinates  the  corresponding  values  of  x  and 
y  are 


% 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

13 

y 

I 

1 

2- 

i 

i 

i 

i 

1 

T 

1 

8" 

i 

A 

tV 

A 

tV 

By  (7) 

=  2.578,  error  .5%; 

By  (8),  ^  =  2.585,  error  .8%; 

By  (9),  ^  =  2.573,  error  .3%; 

By  (10),  A  =  2.572,  error  .3%. 


2.  The  accuracy  of  the  approximation  is  much  increased 
by  taking  the  ordinates  nearer  together,  as  is  shown  by  the 
following  evaluation  of 

h     1+^ 


X' 


The  value  of  this  integral  is  equal  to  the  area  under  the 
curve 

I 


'     i+x' 
from  x=o  ta  x  =  i.     Dividing  the  area  into  twelve  parts  by 

thirteen  equidistant  ordinates  the  value  of  I    is  found  to  be 

Jq    i-\-x 

By    (7),  0.69314866,  error  0.00000148; 

By    (8),  0.69315046,  error  0.00000328; 

By    (9),  0.69314725,  error  0.00000007; 

By  (10),  0.69314722,  error  0.00000004. 

The  correct  value  is,  of  course,  loge  2,  which  is  0.693 14718. 
Formulas  (7)  and  (8)  are  Simpson's  Rules,  (10)  is  Weddle's 
Rule. 


120 


EMPIRICAL  FORMULAS 


3.  Apply  the  above  formulas  to  the  area  of  that  part  of  the 
semi-ellipse  included  between  the  two  perpendiculars  erected  at 
the  middle  points  of  the  semi-major  axes.  Let  this  area  be 
divided  into  twelve  parts  by  equidistant  ordinates. 

Since  the  equation  of  the  ellipse  is 

these  ordinates  are 

Ws  b,   AVT^  b,  IVS  b,    iVYs  b,   iV^s  b,  A>/h3  ^,    b, 
AV^  6,  WJs  b,  W^S  b,  |V8  6,  ^V7^  b,  ^Vl  b. 

By    (7),  ^=0.956609906; 

By    (8),  i4  =0.956608006; 

By    (9),  i4  =0.956611406; 

By  (10),  A  =0.956611406. 

The  correct  value  to  seven 
places  is  0.956611506. 

In  the  application  of  these 
formulas  it  is  highly  desirable 
to  avoid  large  differences  among 
the  ordinates.  For  that  reason 
the  formulas  do  not  give  so 
good  results  when  applied  to 
the  quadrant  of  the  ellipse. 

4.  The  area  under  the  curve, 
Fig.  28,  determined  by  the 
following  sets  of  values: 


3.0 
2.5 
2.0 
1.5 

y 

, 

*N. 

/ 

/ 

N 

^ 

/ 

/ 

0.5 
a 

X  I 


A        .6        .8       1.0      1.2      » 

Fig.  28. 
>  .2  .4 


.6 


i.o 


1.2 


1-5 


2.2 


2.7 


2.6 


2.3 


y   \   i-o 

is  by  (7) 

^=i---5-(i-o+6.o+4.4+io.8+5.24-9.2  +  2.i)  =  2.58, 
and  by  (8), 

^=1-1(1.0+4.54-6.6+5.4+7.8+6.9+2.1)  =  2.5725. 


2.1 


NUMERICAL   INTEGRATION 


121 


X1.2 
ydx. 

The  area  found    is    therefore  the  approximate    value    of    this 
integral 

5.     Find  the  area  under  the  curve  determined  by  the  points 

2.8       3.2       3.6       4.0      4.6       4.8 


XI     1.5       1.9       2.3 


5-0 


y\o       .40     1.08     1.82     2.06     2.20     2.30     2.25     2.00     1.80     1.5 

The  points  located  by  the  above  sets  of  values  are  plotted 
in  Fig.  29  and  a  smooth  curve  drawn  through  them.     The  area 


y                    

^"T~M^ 

2    ■        --  Jr\ 

L                  ~^ 

"2 

IN 

/ 

A 

-V 

7 

^ 

~l 

1 

^              r 

y 

t 

J- 

/^ 
^ 

1 — 1 1 X 

1  2  3  ,4  6 

Fig.  29. 

is  divided  into  strips  each  having  a  width  of  .4.  Rectangles 
are  formed  with  the  same  area  as  the  corresponding  strips. 
The  eye  is  a  very  good  judge  of  the  position  of  the  upper  bound- 
ary of  each  rectangle.  Adding  the  lengths  of  these  rectangles 
and  multiplying  the  sum  by  .4  the  area  is  found  to  be  6.644. 
By  Simpson's  Rule,  formula  (7),  are  found 


for  /?  =  .2, 
/j  =  .4, 


^=6.639, 
^=6.645. 


The  graphical  determination  of  areas  can  be  made  to  3aeld 
a  close  approximation  by  taking  narrow  strips,  and  where  the 
points  are  given  at  irregular  intervals  the  area  can  be  found 
more  rapidly  than  by  the  application  of  Simpson's  Rules. 


122  EMPIRICAL   FORMULAS 

6.  A  gas  expands  from  volume  2  to  volume  10,  so  that  its 
pressure  p  and  volume  v  satisfy  the  equation  pv  =  ioo.  Find 
the  average  pressure  between  v  =  2  and  v  =  io. 

The  average  pressure  is  equal  to  the  work  done  divided  by 

8.     The  work  is  equal  to  the  area  under  the  curve  p  =  ^^  from 

V 

v  =  2  to  2>  =  10,  which  is 

no  jQQ  r      T  ^"^ 

I      — dv=ioo\\ogv\    =160.044. 

J2  V  L  J2 

That  this  area  represents  the  work  done  in  expanding  the 
volume  from  2  to  10  becomes  evident  in  the  following  way. 
Let  5  represent  the  surface  inclosing  the  gas,  ps  will  then  be 
the  total  pressure  on  that  surface.  The  element  of  work  will 
then  be 

dW  =  psdn, 

when  dn  lepresents  the  element  along  the  normal. 

W=Cpsdn. 

But 

sdn  =  dv, 
and 

W=J'pdv. 

This  is  the  equation  above.  The  average  pressure  over  the 
change  of  volume  from  2  to  10  is 

160.944-^8  =  20.118. 

7.  Find  the  mean  value  of  sin^  x  from  x  =  o  to  a:  =  27r.  Plot 
the  curve  y  =  sin^  x  by  the  following  values  of  x  and  y : 


T 

TT 

TT 

TT 

57r 

TT 

0 



— 

— 

— 

— 

12 

6 

4 

3 

12 

2 

0 

.0670 

.2500 

.5000 

.7500 

•9330 

I. COCO 

7^ 

27r 

J^ 

5^ 

IITT 

12 

3 

4 

6 

12 

9330       -7500       .5000       .2500       .0670 


NUMERICAL   INTEGRATION 


123 


X 

IT 

131 
12 

77r 
6 

Si 
4 

47r 
3 

L7Z 
12 

3![ 

2 

y 

o 

.0670 

.2500 

.5000 

.7500 

•9330 

I. 0000 

X 

ig-rr 
12 

Si 
3 

11 
4 

IITT 

6 

235 
12 

27r 

•9330       -7500       .5000       .2500       .0670  o 

Applying  Simpson's  Rule,  formula  (7),  the  area  is  found  to 
be  IT.     The  mean  value  is  the  area  divided  by  27r  or  .5. 

8.  A  body  weighing  100  lb.  moves  along  a  straight  line 
without  rotating,  so  that  its  velocity  v  at  time  t  is  given  by  the 
following  table: 


^  sec 

I 

3 

5 

7 

9 

V  ft./sec 

1-47 

1.58. 

1.67 

1.76 

1.86 

Find  the  mean  value  of  its  kinetic  energy  from  t  =  i  to  t  =  g. 


t 

I 

3 

5 

7 

9 

1)2 

2.1609 

2.4964 

2.7889 

3.0976 

3-4596 

Kinetic  energy . 

3-355 

3.876 

4-331 

4.810 

5-372 

Plotting  kinetic  energy  to  /,  the  area  under  the  curve  is 
34.755.  This  divided  by  8  gives  the  mean  kinetic  energy  as 
4.357- 

Volumes  AV 

Fig.  30  explains  the  ap- 
plication of  the  formulas 
to  the  problem  of  finding 
the  approximate  volume  of 
an  irregular  figure.  Tiie  area 
of  the  sections  at  right  angles 
to  the  axis  of  x  are : 

^2=p(>'6  +  43'9+>'8), 

Az=\k{y2+Ay7-\-y^)'  Fig.  30. 


124  EMPIRICAL  FORMULAS 

If  the  areas  of  these  sections  be  looked  upon  as  ordinates, 
h  being  the  distance  between  two  adjacent  ones,  it  is  evident 
that  the  volume  may  be  represented  by  the  area  under  the 
curve  drawn  through  the  extremities  of  these  ordinates. 

Substituting  the  values  of  Ai,  A2,  and  A3  in  this  equation, 
the  volume  becomes 

V  =  lh[lk{yi-{-4y5+y4.)-\-ik(y6+4y9-\-y8)-\-lk(y2+4y7+y3)] 
=  ihk[yi-\-y2-{-y3-\-yA-\-4{y5+y6+y7-\-ys)  +  i6yQ] 

In  order  to  apply  formulas  (8),  (9)  and  (10),  the  solid  would 
have  to  be  divided  differently,  but  the  method  of  application 
is  at  once  evident  from  the  above  and  needs  no  further  discussion. 

1.  The  following  are  values  of  the  area  in  square  feet  of  the 
cross-section  of  a  railway  cutting  taken  at  intervals  of  6  ft. 
How  many  cubic  feet  of  earth  must  be  removed  in  making  the 
cutting  between  the  two  end  sections  given? 

91,        95,         100,         102,        98,        90,         79. 

These  cross-section  -areas  were  obtained  by  the  application 
of  Simpson's  Rules. 
By  (7), 

F=^- 6(91 -f  380+200 -f  408+196+360 -f  79)  =3428; 

By  (8), 

F  =  f- 6(91 +  285+300+204+294+270+79)  =3426.8. 

2.  ^  is  the  area  of  the  surface  of  the  water  in  a  reservoir 
when  full  to  a  depth  h. 


hit.... 

30 

25 

20 

15 

10 

5 

0 

A  sq.ft. . 

26,700 

22,400 

19,000 

16,500 

14,000 

10,000 

5,000 

NUMERICAL  INTEGRATION  125 

-  Find  (a)  the  volume  of  water  in  the  reservoir,  (b)  the  work 
done  in  pumping  water  out  of  the  reservoir  to  a  height  of  loo  ft. 
above  the  bottom  until  the  remaining  water  has  a  depth  of 
lO  ft. 

7  =  1(26,700+89,600+38,000+66,000+28,000+40,000+5,000) 
=  488,833  cu.  ft. 

Work  =  'Z£;  |      A(ioo  —  h)dh,  where  7£;  =  weight  of  i  cu.ft.  of  water 

Jio 

=  62.3  lb.  The  value  of  this  integral  will  be  approximately  the 
area  under  the  curve  determined  by  the  points 

30  25  20  15  10 


^(100  — /f) -i  1,869,000   1,680,000   1,520,000   1,402,500   1,260,000 

multiplied  by  62.3. 
This  area  is  equal  to 

1(1,869,000+6,920,000+3,040,000+5,610,000  +  1,260,000) 
=  31,165,000. 

Multiplying  this  by  62.3  gives  the  work  equal  to  1,941,579,500 
ft.-lb. 

3.  When  the  curve  in  Fig.  29  revolves  about  the  x-axis, 
find  the  volume  generated. 

The  areas  of  the  cross-sections  corresponding  to  the  given 
values  of  x  are  given  in  the  following  table: 


.4  .0  .6  i.o  1.2 


A 


TT      2.257r       4.847r       7.29^       6.76^       5  •  29^       4.41^ 

By  (7)  F  =  5.8627r=  18.416. 

By  (8)  F  =  5.8o37r  =  18.231. 

4.  When  the  curve  in  Fig.  30  revolves  about  the  x-axis, 
find  the  volume  generated  from  x=i  tox  =  4.2.  From  the  curve 
the  following  sets  of  values  are  obtained : 


126 


EMPIRICAL  FORMULAS 


I.O  1.2 


1.4    1.6    1.8 


2.0 


2.2 


2.4 


2.6 


II 


29   'S3 


87   1.37   I. 71   1.90 


2.01 


y^     o     .012  .084  .281 
X  2.8   3.0   3.2   3 


757  1.877  2.924  3.610  4.040 
4    3-6    3-8    40   4.2 


2 .  06  2.12  2.2 


27   2.30   2.28   2.25   2.20 


f         4.2444.4944.84    5.153     5.290     5.198     5.062    4.84 
The  volume  is  by  (7) 

7r-i'i(i49.oo4)=3i.2  cu.  units. 

Centroids 
Let  the  coordinates  of  the  centroid  of  an  area  be  represented 
by  X  and  y.    Then  from  the  calculus 

I    xydx 


I    ydx 

-  I    y'^dx 
ydx 


y= 


The  integral  in  the  numerator  of  the  value  of  x  may  be 
represented  by  the  area  bounded  by  the  curve  Y  =  xy,  the  x-axis 
and  the  two  ordinates  x  =  a  and  x  =  h.  The  original  area  is 
bounded  by  the  curve  whose  ordinates  are  represented  by  y, 
the  X-axis  and  the  two  ordinates  x  =  a  and  x  =  h.  The  integral 
in  the  numerator  of  the  value  of  y  may  be  represented  by  the 
area  bounded  by  the  curve  Y=y'^,  the  x-axis  and  the  two 
ordinates  x  =  a  and  x  =  h. 

For  a  voliune  generated  by  revolving  a  given  area  about  the 
X-axis 

IT    I     ^'^Xfi^X 


x  = 


'X 


y'^dx 


NUMERICAL  INTEGRATION 

When  the  volume  is  irregular 


127 


Axdx 


i 


Adx 


The  process  of  finding  the  coordinates  of  the  centroid  of  the 
area  in  Fig.  28  is  shown  in  the  table: 


X 

y 

xy  • 

y2 


0 

.2 

•4 

.6 

.8 

1.0 

I.O 

I-S 

2.2 

2.7 

2.6 

2.3 

0.00 

0.30 

0.88 

1.62 

2.08 

2.30 

1. 00 

2.25 

4.84 

7.29 

6.76 

529 

0.000 

0.450 

1.936 

4-374 

5.408 

5.290 

1.2 

2.1 
2.52 
4.41 
5.292 


The  area  under  the  curve  Y=xy  is 

^[0.00+1.20+1.76-1-6.48+4.16+9.20+2.52]  =  1.688; 

-     1.688      ^ 
2.58 

The  area  under  the  curve  F  =  J>^  is 

^[1.00+9.00+9.68+29.16+13.52  +  21.16+4.41]  =  2.931 

-_2-93i 


2.58 


1-136. 


As  was  pointed  out  before,  large  changes  in  the  ordinates 
must  be  avoided. 

For  the  volume  generated  by  revolving  the  area  about 
the  x-SLxis 

-_7rTg-[o.ooo+ 1.800+3.872 +  17.496+ 10.816+ 21. 160+5. 292] 
7rxF[i.oo+9.oo+9.68+29.i6+i3.52+2i.i6+4.4i] 

60.436 


87-93 


.687. 


128 


EMPIRICAL   FORMULAS 


Moments  of  Ineriia 

The  expression  for  the  moment  of  inertia  of  an  area  about 
the  ^'-axis  is 

/y=  j  x^ydx. 

About  the  ac-axis 

/x=  j    x-fdy. 

When  the  equation  of  the  curve  is  known  these  integrals 
can  be  calculated  at  once,  but  when  this  is  not  the  case  approxi- 
mate methods  must  be  resorted  to. 

I.  The  process  of  finding  the  approximate  values  of  these 
integrals  is  shown  in  the  table  below.  The  values  of  x  and  y 
are  taken  from  Fig.  28. 


X 

y 

x'^y 

\y' 


0 

.2 

•  4 

.6 

.8 

1.0 

I.O 

1-5 

2.2 

2.7 

2.6 

2.3 

O.CXX) 

0.060 

0.33s 

0.972 

1.664 

2.300 

0.333 

I. 125 

3  549 

6.561 

5.859 

4.056 

I.  2 
2.1 
3.024 

3  087 


If  the  values  of  x^y  be  plotted  to  x  we  will  have  a  curve 
under  which  the  area  represents  the  moment  of  inertia  of  the 
area  in  Fig.  28  about  the  y-3ixis. 

Dividing  this  by  the  area  found  before,  there  results  for  the 
radius  of  gyration  about  the  3;-axis 


i2,2  =  .526. 

Plotting  iy^  to  X  and  finding  the  area  under  the  curve  so 
determined 

7^=4.6x36, 
and 

jRx2  =  1.788. 


NUMERICAL   INTEGRATION 


129 


2.  The  form  of  a  quarter  section  of  a  hollow  pillar,  Fig.  31, 
is  given  by  the  following  table.  Find  the  moment  of  inertia 
of  the  section  about  the  axes  of  x  and  y. 


y 
.5 

n 

■-^ 

r-^ 

A 

"^ 

\ 

IN 

N 

I 

\ 

.3 

N 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

.i 

\ 

\ 

Fig.  31. 


X 

Y 

xW     ■ 

y 

X 

y^X 

00 

050 

.00000 

00 

100 

. 00000 

05 

OSS 

.00014 

OS 

108 

.00027 

10 

068 

.00068 

10 

116 

.00116 

IS 

078 

.00175 

IS  . 

120 

.00270 

20 

096 

.00384 

20 

125 

.00500 

25 

116 

.00725 

25 

130 

.00812 

30 

148 

.01332 

30 

133 

.01197 

3S 

200 

.02450 

35 

140 

.01715 

40 

300 

. 04800 

40 

150 

.02400 

45 

21S 

•043S4 

45 

215 

•04354 

SO 

000 

. 00000 

SO 

000 

.00000 

In  the  above  table  X  stands  for  the  width  of  the  area  parallel 
to  the  X-axis  and  Y  for  the  width  parallel  to  the  ^/-axis.  The 
area  is  0.066. 

The  moment  of  inertia  about  the  ^'-axis  is 


X 


x'^Ydx  =  .oo']^6\ 


i?/  =  :2?M  =  o.iii5. 

.000 


130  EMPIRICAL   FORMULAS 

The  moment  of  inertia  about  the  ^-axis  is 

I    y^Xdy  =  .oo6ig; 
Jo 

^'^  =  ^'  =  -^38, 

where  R  stands  for  the  radius  of  gyration. 

The  values  of  the  above  integrals  were  computed  by  for- 
mula (7). 


APPENDIX 


If  a  chart  could  be  constructed  with  all  the  different  forms 
of  curves  together  with  their  equations  which  may  arise  in 
representing  different  sets  of  data  it  would  be  a  comparatively 
simple  matter  to  select  from  the  curves  so  constructed  the  one 
best  suited  for  any  particular  set.  Useful  as  such  a  chart  would 
be  its  construction  is  clearly  out  of  the  question.  The  most 
that  can  be  done  of  such  a  nature  is  to  draw  a  number  of  curves 
represented  by  each  one  of  the  simpler  equations. 

A  word  of  caution  is,  however,  necessary  here.  A  particular 
curve  may  seem  to  the  eye  to  be  the  one  best  suited  for  a  given 
set  of  data,  and  yet,  when  the  test  is  applied,  it  may  be  found 
to  be  a  very  poor  fit.  It  is  of  some  aid,  nevertheless,  to  have 
before  the  eye  a  few  of  the  curves  represented  by  a  given  formula. 

The  purpose  of  the  following  figures  is  to  illustrate  the 
changes  in  the  form  of  curves  produced  by  slight  changes  in 
the  constants.  Figs.  I,  II,  III,  and  IV  show  changes  produced 
by  the  addition  of  terms.  Figs.  V  to  XIX  changes  in  form 
produced  by  changes  in  the  values  of  the  constants,  and  Fig. 
XX  the  changes  in  form  brought  about  by  varying  both  the 
values  of  the  constants  and  the  number  of  terms. 

A  discussion  of  all  the  figures  is  unnecessary.  A  few  words 
in  regard  to  one  will  suffice.  Formula  XIV,  for  example, 
y  =  a-\-bx'',  an  equation  which  can  be  made  to  express  fairly 
well  the  quantity  of  water  flowing  in  many  streams  if  x 
stands  for  mean  depth  and  y  for  the  discharge  per  second, 
represents  a  family  of  triply  infinite  number  of  curves.  Fixing 
the  values  of  b  and  c  and  varying  the  value  of  a  does  not 
change  the  form  of  the  curve,  but  only  moves  it  up  or  down 

131 


132 


EMPIRICAL   FORMULAS 


along  the  >'-axis.  Keeping  the  values  of  a  and  b  constant  and 
varying  the  value  of  c,  the  formula  will  represent  an  infinite 
number  of  curves  all  cutting  the  ^'-axis  in  the  same  point.  In 
the  same  way,  keeping  the  value  of  a  and  c  constant  and  vary- 
ing the  value  of  b,  an  infinite  number  of  curves  is  obtained, 
all  of  which  cut  the  >;-axis  in  a  fixed  point.  In  Fig.  XIV  the 
quantity  a  is  constant  and  equal  to  unity,  while  b  and  c  vary. 

To  one  trained  in  the  theory  of  curves  the  illustrations  are, 
of  course,  of  no  essential  value,  but  to  one  not  so  trained  they 
may  be  of  considerable  help. 

The  text  should  be  consulted  in  connection  with  the  curves 
in  any  figure.  The  figures  are  designated  to  correspond  to  the 
formulas  discussed  in  the  first  five  chapters. 


V 

/ 

/ 

// 

/' 

/ 

^ 

^ 

^ 

^ 

^ 

^ 

=5§ 

M 

■    " 

.^ 

"^ 

^ 

^ 

^ 

S 

\ 

\ 

^ 

^ 

=^ 

^ 

^v 

-0  4 

\ 

\ 

\^ 

-OJ 

\ 

4        S        6        ; 

Fig.  I. 


(1)  y=l-.lx 

(2)  j/=l-.lx+.01x2 

(3)  j/=l-.lx+.01x2-.001xS 

(4)  y=l-.lx+.01x2-.001x3-f 

.OOOlx* 

(5)  ?/=l-.lx+.01x2-.001x'+ 

.0001x*-.00001x^ 

(6)  y=-l-.lx+.01x2-.001x'+ 

.0001x*-.00001x*+,000001x8 

See  formula  I,  page  13 


APPENDIX 


133 


V 
1.4 

ll 

1.2 

^h 



1 

l\<2> 

-«!- 

=» 

--. 

06 

€C 

^ 

0.4 
0.2 
0 

f 

/ 

// 

f 

-0.2 

i 

-0.4 
-0.6 
-0.8 

\j 

-1, 



(1)  y=l-l/x 

(2)  y=l-l/x+l/x2 

(3)  y=l-l/x+l/x^-l/x^ 

(4)  y=l_l/a;+l/a;2-l/a;'+l/x* 

(5)  y=l-l/x+l/x^-l/x^+ 

l/x*-l/x^ 

(6)  y=l-l/x+lA2-l/x3+ 

l/x*-l/xHl/x8 

See  formula  II,  page  22 


Fig.  II. 


2.8 

V- 

/ 

1 

/ 

/ 

/ 

/ 

/ 

2.2 

2 

1.8 

1.6 

7 

I 

/ 

y 

i 

/ 

// 

' 

/ 

/ 

/; 

/ 

/J 

^ 

^ 

-Hi 

2r- 

N, 

1.4 

— , 

y 

V 

^ 

^ 

X 

^ 

s^ 

1.2 

a 

0.8 
0.6 

oa 

0.2 
0 

^ 

^ 

^ 

\ 

s\ 

N 

5 

J 

t 

J 

3 

K 

J 

J      1 

0       1 

1     1 

d      x 

(1)  -=l-.la; 

y 

(2)  -=l-.lx+.01x2 
LV 

l3)   -=1-.1x+.01x2-.001x3 

y 

(4)  -=l-.lx+.01x2-.001..3^ 

y 

.000 Ix* 

(5)  -=l-.lx+.01x2-.00u3+ 

y 

.0001x*-.00001x5 

(6)  -  =  r-.lx+.01x2-.001x3+ 

y 

.OOOlx^-.OOOOlx^+.OOOOOlx® 
See  formula  III,  page  25 


Fig.  III. 


134 


EMPIRICAL   FORMULAS 


A 

^ 

,y^ 

'^ 

^=* 

^ 

^ 

^ 

J2- 

^ 

^ 

""^ 

■^ 

^ 

=-; 

^^ 

<4) 

> 

^ 

^ 

^ 

N 

^ 

^ 

ll 

//I 

^ 

(1)  y«=l-.lx 

(2)  i/J=l-.ix+.oix» 

(3)  i/«=l-.lx+.01i»-.001z» 

(4)  v«=l-.lx+.01i«-.001x«+ 

.000 lx< 

(5)  !/«=!-. Ix+.01x2-.001x'+ 

.0001i*-.00001x* 

(0)   j/2=l-.lx+.01x2-.001x'+ 

.0001x<-.00001x''+.000001i« 

See  formula  IV,  page  25 


5        6        7        8        9       10      U       02 

Fig.  IV. 


s 

/ 

/ 

"" 

1 

/ 

/ 

2.8 
14 

z 

1 

/ 

'^ 

/ 

/ 

/ 

/ 

/ 

f 

^ 

/ 

/ 

^ 

^ 

/ 

/ 

^ 

^ 

y 

y 

-> 

^ 

IJ 

2 

y 

y 

^ 

^ 

__ 



^ 



— 

— 

r~ 

— 

r- 

v~ 

— 

j^ 

— 

— 

— 



^ 

^^ 

- 

^ 

(^ 

1 — 

\ 

^V 

\ 

^ 

""^ 

^ 

^ 

— 

— 

\ 

s: 

■^ 

^«. 

■^ 

■ 

^ 

-^ 

. 

0 

^ 

^ 

^ 

^ 

im 

^ 

— 

— 



(1)  l/=(.5)^ 

(2)  i/=(.6)* 

(3)  2/=  (.7)^ 

(4)  j/=(.8)* 

(5)  2/=  (.9)* 

(6)  y=(.95)* 

(7)  i/=.99)* 

(8)  2/=(1.01)' 

(9)  |/=(1.05)* 

(10)  j/=(l.l)* 

(11)  2/=(1.2)* 

See  formula,  V,  page  27 


Fig.  V. 


APPENDIX 


135 


V 
2 

"> 

2>^ 

;;^ 

-^ 

z::z 

■ 

, 

/ 

K 

(4)^ 

^ 

^ 

-^ 

1.4 

/ 

'// 

/ 
^ 

^ 
^ 

^ 

(6),^ 

^- 

/^ 

y. 

^ 

(7) 

92 

-— 

1 

i 

^ 

, 



N 

?^ 

— 

:::: 

,(11) 



0.8 

as 

'v.i 

:r^ 

-^ 

(12) 

^ 

' 

\ 

^ 

"l^ 

\ 

(W8 

? 
-0.2 

-d.1 

\ 

\ 

^\ 

\ 

\ 

\ 

N 

\ 

\ 

— \ 

\ 

% 

c 

\ 

£ 

« 

t 

5 

1 

)       L 

I       1 

I       X 

(1)  2/  =  2-(.5)* 

(2)  2/  =  2-(.6)* 

(3)  y=2-(.7)* 

(4)  j/=2-(.8)* 

(5)  y-2-(.85)* 

(6)  j/=2-(.9)'»^ 

(7)  2/  =  2-(.95)* 

(8)  y=2-(.97)* 

(9)  j/=2-(.99)* 

(10)  2/=2-(1.01)* 

(11)  y=2-(1.03)* 

(12)  y=2-(1.05)* 
(13)y=2-(1.07)* 
(14)  j/=2-(1.08)* 

See  formula  VI,  page  28 


Fig.  VI. 


V 

^ 

^ 

:=== 

^ 

^ 

(\y 

<y 

y 

^ 

■^ 

^^ 

/ 

^/ 

> 

y 

y^ 

^ 

// 

v 

/ 

.jy 

^ 

^/ 

y 

(5)^ 

-^ 



Y/ 

^ 

(6) 

.^ 

■ — ' 

^ 

_(7); 

— ■ 

— 

>s 

^^ 

— 

— 



\^ 

\^ 

"-^ 

s^ 

\(12 

^^ 

^ 

__ 

■ 

. 

. 

L 

% 

( 

5 

5 

r 

3 

9       1 

0        1 

1        1 

i     X 

(1)  logj/=l-.5(.5)^ 

(2)  log2/=l-.5(.6)^ 

(3)  log2/=l-.5(.7)* 

(4)  Iog2/=l-.5(.8)^ 

(5)  Iogy=l-.5(.9)* 

(6)  log2/=l-.5(.95)* 

(7)  Iogj/=l-.5(.98)'^ 

(8)  logj/=l-.5(1.02)* 

(9)  logy=l-.5(l.l)* 

(10)  logy=l-.5(1.2)3J 

(11)  logy=l-.5(1.3)aJ 

(12)  log2/  =  l-.5(1.5)a' 

(13)  log2/=l-.5(2)a^ 
base=  10 

See  formula  VII,  page  32 


Fig.  VII. 


136 


EMPIRICAL  FORMULAS 


» 

* 

^ 

w 

— 

u 

(» 

^ 

^ 

■J22. 

.^ 

■ — 



> 

S 

t 

-^ 

"^ 

^ 

■^ 

N 

\ 

^ 

Vw. 

^ 

^-- 

r 

\ 

.^ 

s. 

^ 

^ 

-^ 

^ 

X 

N 

X 

N 

^ 

^ 

\ 

N 

X 

N, 

0  1 

\ 

S 

N 

\ 

N 

\ 

\ 

>a4 

\ 

(1)  i/=2-.01i-(.5)* 

(2)  v  =  2-.03a!-(.5)» 

(3)  v=2-.05x-(.5)* 

(4)  v=2-.08x-(.5)* 

(5)  v=2-.lx-(.5)* 

(6)  v=2-.12x-(.5)* 

(7)  i/=2-.15i-(.5)* 

(8)  i/=2-.2x  -(.5)* 

See  formiila  VIII,  page 


2        3        4         5        6        7 

Fig.  VIII 


» 

\ 

1 

\ 

\ 

\ 

\ 

/ 

\\ 

// 

\ 

\ 

/ 

/ 

y 

\ 

\ 

/ 

/ 

Vi> 

\\ 

V 

/ 

t 

^ 

^ 

■^ 

^ 

^ 

= 

^ 

^ 

\ 

-^ 

I     i 

i 

i 

I 

i        t 

i 

r 

J 

»     1 

0       1 

i     i, 

2    m 

(1)  |/=lO-81-.3&r  +  .03x» 

(2)  j/=  10-54 -.24x  +  . 02x2 

(3)  y=10-27-.12x  +  .01x2 

(4)  y= 10.135 -.06x  +  .005x» 

(5)  2/=10--135  +  .06x-  005x2 

(6)  y=10--54  +  .24x-.02x2 

See  formula  IX,  page  37 


Fig.  IX. 


APPENDIX 


137 


V. 

\ 

\ 

J' 

\ 

w. 

15^ 

1.4 

\ 

^ 

^:=5 

:r^ 

^ 

(3) 

V- 

-— 

""^ 

\ 

/     \ 

1 

V 

y 

N 

*s 

"/ 

/ 

nn 

^ 

^ 

y. 

.y 

:: — ■ 

(6) 

5 

i 

6 

r 

s 

9        1 

0        1 

1        J 

i       * 

(1)  i/=(1.01)*  (1.05)(l-2)^ 

(2)  j/=(1.01)*  (1.05)(1-16)^ 

(3)  y=(1.01)*  (1.05)(1-15)^ 

(4)  y=(.5)==  (2)  (1-24)^ 

(5)  y=(.5)*  (2) (1-23)^ 

(6)  y=(.5)*  (2) (1-2)'' 

See  formula  X,  page  37 


Fig.  X. 


/ 

\ 

/ 

\ 

\ 

/ 

\ 

i 

/ 

"\ 

\^ 

1 

,^ 

^ 

\, 

\ 

<^J 

r 

\ 

\ 

k^ 

\ 

V 

L 

>. 

\^. 

\ 

N 

\ 

^ 

/- 

\ 

v^ 

\ 

^ 

.^^ 

^ 

.^^ 

N 

3u 

::^ 

~— 



^ 

— 

■~^ — 

■ — 







•I 

4 

5 

G 

8 

S 

1 

)        1 

L       L 

J      » 

(1)  l/= 


(2)   J/= 


(3)  y- 


(4)  y= 


(5)   2/ 


.2- 

.li+.05i2 

X 

2— 

.lx+.07x2 

X 

.2- 

.lx+.la;2 

x 

.2- 

.la;+.2x2 

X 

.2-.lx+.4x2 
See  formula  XI,  page  38 


Fig.  XI. 


138 


EMPIRICAL  rORMULAS 


V  - 

y 

y 

^ 

¥ 

y 

\\ 

/ 

-. — 

—■ 

' 

•1 

/ 

. 

y^'' 

3- 

// 

^ 

-IIL 

i 

f     \ 

V^ 

-!2L. 



__ 

:( 

^ 

__ 

— 

oi— 

1     i 

X 

i 

\      ( 

\ 

J 

1 

J     1 

1     1 

2        <B 

(1)  X/-5X-* 

(2)  y=5x-2 

(3)  y-5x-l 

(4)  y=5x--l 

(5)  i/=5x--2 

(6)  i/=5x--'* 

See  formula  XII,  page  42 


Fig.  XII. 


. — 

% 

"^ 

^ 

:^ 

1 

H 

^ 

__ 

I 

y 

^ 

-^ 

\ 

/ 

/ 

^ 

--- 

--^ 

\ 

^ 

' 

lO, 

" 

1 

V 

- 

, 

-^ 

I 

' 

■  ■ 

\ 

— 

(.">) 

(l)y=H-Iogx+.llog2x; 

V=— 1.5  (min.)  when  log  x=  —5 

(2)  y=l+log  x+.011og2  x; 

j/= — 24  (min.)  when  log  x=  —50 

(3)  |/=l+.2  1ogx+.31og2x 

(4)  2/=l— log  x+log*  X 

(5)  j/=l-logx+.5  log*x 

See  formula  XIII,  page  44 


Fig.  Xm. 


APPENDIX 


139 


y 

^ 

.^ 

^^ 

:rj 

^ 

^ 

=^ 

^ 

-~^ 

(4) 

<; 

^(S) 

^ 

^ 

N 

^ 

N^ 

\ 

I     i 

i 

i 

. 

>     ( 

r 

J 

)     1 

0       1 

1  1 

2      '» 

(1)  |/=H-.008xl-7 

(2)  2/=l+.007a;l-6 

(3)  |/=H-.006xl-6 

(4)  y=l-. 002x2 

(5)  y=l-. 003x2-1 

(6)  2/=  1-. 004x2-2 

See  formula  XIV,  page  45 


Fig.  XIV. 


^ 

\ 

^ 

\ 

\ 

\ 

"\ 

^ 

\ 

\ 

\ 

^ 

^ 

\ 

\ 

\ 

s. 

\. 

^ 

K 

\ 

N, 

\^ 

\ 

r\ 

^ 

^ 

~>^, 

^ 

\ 

s. 

^ 

<v. 

\^ 

^ 

^ 

i 

i 

4 

S 

c 

J 

8 

( 

1 

)       1 

I    .  L 

i"& 

(1)  y=(2.0)  10 --01^^ 

(2)  2/=  (1.6)  10- •02x1-7 

(3)  J/=(1.2)  10--03il-^ 

(4)  y=(1.0)  10 -•04x1-36 

(5)  y=(0.8)  10- -05x1-24 

(6)  2/=  (0.6)  10 -•06x1-12 

See  formula  XV,  page  49 


Fig.  XV. 


140 


EMPIRICAL  FORMULAS 


V  — 

s 

r^ 

^ 

% 

^ 

^ 

^ 

w 

7\ 

\1 

s 

\, 

< 

\ 

'^V 

V 

N 

^ 

^ 

^ 

K 

-^ 

- 

"^ 

•^ 

i 

s 

\ 

I 

« 

i 

« 

y    1 

0       1 

1 

e     X 

(1)  (1/-2)  (xf.S) 1 

(2)  (v-2)  (X+.75) 1.5 

(3)  (1/-2)  (x+l)  =  -2 

(4)  (1/+.1)  (x+4)=8.2 

(5)  (v+.l)  (x+3)  =  6.3 

(6)  (I/+.1)  (x+2)=4.2 

See  formula  XVI,  page  53 


Fig.  XVI. 


'       '       (1)  y=h  10a;+24 


(2)  y  =  \  101-24 

2 

(3)  i/=i  10a;+2 

(4)  2/=To  10^ 

■5 

(5)  y  =  -lQX  +  l 

See  formula  XVIo,  page  56 


5      0      7       8 

Fig.  XVIa. 


9       10      U      12      » 


APPENDIX 


141 


(1)  y=.5e-0lx+e-05x 

(2)  y=2e-05a;_.5e.la; 

(3)  y=2.25e-05a;_.75e.l* 

(4)  y=1.8e-01a;_.3e.lx 

(5)  y=1.92e--lar_.42e-.01a; 

(6)  y=2e--05a;_e-.01x 

(7)  y=4.2e--2a:-3.5e--25a; 

(8)  y=4.5e--2a;-4.1e--25a: 

(9)  2/=.25e--01a;_.]3e--15x 

(10)  y=e-  l«-l.le-.2a: 

(11)  y=.27e--01^-.77e--25a; 

(12)  y=e- -1^-26 --253; 

See  formula  XVII,  page  58 


Fig.  XVII. 


p^ 

\ 

-^, 

\^ 

\ 

^ 

\ 

U2) 

\ 

N 

y 

(3) 

N 

\ 

(5) 

— 

,_ 

-J 

^\ 

/ 

^^ 

-^ 

■N 

s 

/- 

^^'^ 

(6)/ 

' 

X 

'^ 

x; 

:\ 

\, 

s/ 

'' 

; 

/|-^ 

■^ 

,^ 

\: 

A 

V 

/ 

\ 

^ 

/ 

^~-. 

--_ 



^ 

(1)  y=e-01a;(1.5  cos  .lx-.5  sin  Ax) 

(2)  2/=e  — •2a;(1.5cos.5x— .5sin.5x) 

(3)  2/  =  e--la:(.6co3  .lx+.8sin  .Iz) 

(4)  y=e^^{.2  cos  .3x-.l  sin  .3a;) 

(5)  y=e-02a:(.4cos.l6x+.17sin.l6x) 

(6)  j/-=.5e~'l^  sin  X 

See  formula  XVIII,  page  61 


0        10       U       12 


Fig.  XVIII 


142 


EMPIRICAL  FORMULAS 


» 

^ 

^ 

/ 

y 

N 

/ 

/ 

V>)^ 

/ 

/ 

Ul_ 

I 

^_^ 

^ 

7^ 

^ 

E— : 

5 

:=« 

^^ 

^ 

— 

■= 

/ 

5 

^^ 

-^ 

t 

/ 

^ 

(3) 

, 

a* 

A 

V' 

^ 

"^ 

\ 

\ 

{ 

< 

I 

i 

I 

0       1 

1  1 

8      ^ 

(1)  i/=2x-l-a;-2 

(2)  t^=3x-5-2.2x-6 

(3)  v-2.3x-8-2x-85 

(4)  v=.lx-l+.5x-2 

(5)  i/=.33x-.0012x3   4- 

(6)  i/=.25x-5+.05x-8 

See  formula  XIX,  page  65 


Fig.  XIX. 


"J 

— 

"H 

-/ 

/-I 

^ 

\ 

, 

— 

f 

r 

^ 

^ 

— ^ 

A 

V 

s 

^ 

/ 

/\ 

s^ 

\ 

X 

N, 

^ 

, — ■ 

^ 

1/ 

^ 

■^ 

— 

\. 

r^ 

^ 

■ — 

— 

1.       S        8        i        5        e        7        8        0       10      11       U     "0 

1 

Pig. 

XI 

Xa. 

(1)  2/=15xl-5(.4)* 

(2)  2/=3x2(.5)*       ^' 

(3)  y=3x-2(1.5)* 

(4)  |/=-.5xl-5(.75)* 

See  formula  XlXa,  page  72 


APPENDIX 


143 


200 


190 


170 


150 


100 


^ 

(2) 

rs 

^ 

'^ 

\ 



7^ 

?-^ 

,-^ 

(3) 

^^ 

/ 

^ 

> 

^i 

^^ 

^ 

N. 

fe 

pN. 

\^ 

"^ 

1 

h 

^ 

'^ 

~^-i*2 

y 

S 

\ 

N 

\ 

^ 

Nj 

^ 

Vi 

^ 

N 

^. 

-^ 

^ 

s^ 

y 

? 


Fig.  XX. 


(1)  2/=166.25-14.5  cos  x-2.75  cos  2x— 10  sin  x 

(2)  y=167.83-20  cos  x-4.33  cos  2x+5.5  cos  3x-13.28  sin  x-17.32  sin  2x 

(3)  y=167.62-17.5  cos  x-2.75  cos  2x+3  cos  3x-1.38  cos  4x-12.42  sin  x-18  sin  2x- 

2.42  sin  3x 

(4)  j/=  167.08-17.22  cos  x-3. 5  cos  2x+5.5  cos  3x-0.83  cos  4x-2.78  cos  5x+ 

0.75  cos  6X-12.14  sin  x-19.05  sin  2x-sin  3x- 1.73  sin  4x+ 1.14  sin  5x 


See  formula  XX,  page  74 


INDEX 


Approximation,  Accuracy  of,  114,  118, 

119,  121 
Area,  11,  114,  121,  124,  125,  126,  128 

Graphical  Determination  of,  121 
Arithmetical  Series,  11,  13,  15,  19,  22, 

25.  27,  28, 32, 33, 37, 38, 44, 58,  61, 104 

Centroids,  ii,  126,  127 
Check,  46,  92 

Common  Difference,  ii,  16 
Compound  Interest  Law,  27 
Constants,  11,  92,  94,  131,  132 

Graphical  Determination  of,  13,  15, 
21,  23,  24,  25,  28,  31,  32,  36,  40, 
44,  47,  48,  51,  55,  60,  64,  65,  67  ,70 

Changes  in,  131 
Curves,  10,  11,  114,  121,  122,  131,  132 

Damped  Vibration,  65 
Derivative,  95 

First,  108 

Second,  108 
Differences,  11,   12,  16,  33,   loi,   115, 
116 

First  Order,  12,  18 

Higher  Order,  12,  16,  18,  19,  22,  loi 
Differential  Equation,  68,  71,  11 1 
Differentiation  of  Tabulated  Functions, 

100,  108 

Elastic  Limit,  14 

Empirical  Formulas,  9,  11,  68,  90,  100 
Errors  of  Observation,  11,  68 
Fourier  Series,  74 

6-ordinate  Scheme,  74,  76 
8-ordinate  Scheme,  78 

lo-ordinate  Scheme,  80 

i2-ordinate  Scheme,  81,  86 

1 6-ordinate  Scheme,  82 

20-ordinate  Scheme,  84 

24-ordinate  Scheme,  85 


Gauss,  107 

Geometrical  series,  11,  15,   29,  42,  45, 
49*65 

Indicator  Diagram,  114 

Integral,  Definite,  114,  118,  121,  125 

Integration,  Numerical,  114,  115 

3  Ordinates,  116 

4  Ordinates,  117 

5  Ordinates,  117 
7  Ordinates,  117 

Interpolation,  11, 100,  loi,  112, 114, 115 
Trigonometric,  107 

Lagrange's  Theorem,  106,  in 

Maximum,  113 

Mechanical  Quadrature,  114 

Method  of  Least  Squares,  11,  19,  90, 91, 

94 
Minimum,  113 
Moment  of  Inertia,  11,  128,  129,  130 

Normal  Equations,  92,  96,  99 

Observation  Equations,  91,  96,  97,  98 

Ratio,  II,  46 
Residuals,  Squares  of,  94 
Period,'  74,  88,  in 
Less  than  27r,  74,  88 

Physical  Basis,  9 

Physical  Law,  9 

Planimeter,  114 

Pressure,  Upward,  of  Water,  49 

Simpson's  Rule,  119,  121,  123,  124 

Taylor's  Theorem,  94,  97 

Volume,  II,  123,  125,  127 

Weddle's  Rule,  119 


144 


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